1. MATH 110 Sec 13-4 Lecture: Expected Value The value of items along with the probabilities that they will be stolen over the next year are shown. What can the insurance company expect to pay in claims on your policy? Is $100 a fair premium for this policy?

2. MATH 110 Sec 13-4 Lecture: Expected Value The value of items along with the probabilities that they will be stolen over the next year are shown. What can the insurance company expect to pay in claims on your policy? Is $100 a fair premium for this policy? For example, the $2000 laptop has a probability of 0.02 of being stolen.

3. MATH 110 Sec 13-4 Lecture: Expected Value The value of items along with the probabilities that they will be stolen over the next year are shown. What can the insurance company expect to pay in claims on your policy? Is $100 a fair premium for this policy? For example, the $2000 laptop has a probability of 0.02 of being stolen. On average, the company will pay out $2000 due to a stolen computer 2% (0.02) of the time.

4. MATH 110 Sec 13-4 Lecture: Expected Value The value of items along with the probabilities that they will be stolen over the next year are shown. What can the insurance company expect to pay in claims on your policy? Is $100 a fair premium for this policy? For example, the $2000 laptop has a probability of 0.02 of being stolen. On average, the company will pay out $2000 due to a stolen computer 2% (0.02) of the time. We quantify this idea by saying that by insuring this computer, the company has an expected payout of $2000 x 0.02 = $40.

5. MATH 110 Sec 13-4 Lecture: Expected Value The value of items along with the probabilities that they will be stolen over the next year are shown. What can the insurance company expect to pay in claims on your policy? Is $100 a fair premium for this policy? Let’s add an expected payout column to our table and then calculate the expected payout for all of the items listed.

6. MATH 110 Sec 13-4 Lecture: Expected Value The value of items along with the probabilities that they will be stolen over the next year are shown. What can the insurance company expect to pay in claims on your policy? Is $100 a fair premium for this policy? This is the expected payout that we just calculated.

7. MATH 110 Sec 13-4 Lecture: Expected Value The value of items along with the probabilities that they will be stolen over the next year are shown. What can the insurance company expect to pay in claims on your policy? Is $100 a fair premium for this policy? This is the expected payout for the $400 iPhone with a 3% (0.03) probability of being stolen.

8. MATH 110 Sec 13-4 Lecture: Expected Value The value of items along with the probabilities that they will be stolen over the next year are shown. What can the insurance company expect to pay in claims on your policy? Is $100 a fair premium for this policy? This is the expected payout for the $600 trail bike with a 1% (0.01) probability of being stolen.

9. MATH 110 Sec 13-4 Lecture: Expected Value The value of items along with the probabilities that they will be stolen over the next year are shown. What can the insurance company expect to pay in claims on your policy? Is $100 a fair premium for this policy? This is the expected payout for the $800 textbooks with a 1% (0.01) probability of being stolen.

10. MATH 110 Sec 13-4 Lecture: Expected Value The value of items along with the probabilities that they will be stolen over the next year are shown. What can the insurance company expect to pay in claims on your policy? Is $100 a fair premium for this policy? Add up all the expected payouts to get: $40 + $12 + $6 + $32 = $90.

11. MATH 110 Sec 13-4 Lecture: Expected Value The value of items along with the probabilities that they will be stolen over the next year are shown. What can the insurance company expect to pay in claims on your policy? Is $100 a fair premium for this policy? Add up all the expected payouts to get: $40 + $12 + $6 + $32 = $90. So, does this premium seem fair?

12. MATH 110 Sec 13-4 Lecture: Expected Value Add up all the expected payouts to get: $40 + $12 + $6 + $32 = $90. A $10 markup on a $90 expected payout is about 11% and seems fair. The value of items along with the probabilities that they will be stolen over the next year are shown. What can the insurance company expect to pay in claims on your policy? Is $100 a fair premium for this policy? So, does this premium seem fair?

13. The value of items along with the probabilities that they will be stolen over the next year are shown. What can the insurance company expect to pay in claims on your policy? Is $100 a fair premium for this policy? MATH 110 Sec 13-4 Lecture: Expected Value Add up all the expected payouts to get: $40 + $12 + $6 + $32 = $90. Let’s look more closely at the calculation giving a $90 expected payout.

14. The value of items along with the probabilities that they will be stolen over the next year are shown. What can the insurance company expect to pay in claims on your policy? Is $100 a fair premium for this policy? MATH 110 Sec 13-4 Lecture: Expected Value Add up all the expected payouts to get: $40 + $12 + $6 + $32 = $90. Let’s look more closely at the calculation giving a $90 expected payout.

15. The value of items along with the probabilities that they will be stolen over the next year are shown. What can the insurance company expect to pay in claims on your policy? Is $100 a fair premium for this policy? MATH 110 Sec 13-4 Lecture: Expected Value Add up all the expected payouts to get: $40 + $12 + $6 + $32 = $90. Let’s look more closely at the calculation giving a $90 expected payout.

16. The value of items along with the probabilities that they will be stolen over the next year are shown. What can the insurance company expect to pay in claims on your policy? Is $100 a fair premium for this policy? MATH 110 Sec 13-4 Lecture: Expected Value Add up all the expected payouts to get: $40 + $12 + $6 + $32 = $90. Let’s look more closely at the calculation giving a $90 expected payout.

17. The value of items along with the probabilities that they will be stolen over the next year are shown. What can the insurance company expect to pay in claims on your policy? Is $100 a fair premium for this policy? MATH 110 Sec 13-4 Lecture: Expected Value Add up all the expected payouts to get: $40 + $12 + $6 + $32 = $90. Let’s look more closely at the calculation giving a $90 expected payout.

18. The value of items along with the probabilities that they will be stolen over the next year are shown. What can the insurance company expect to pay in claims on your policy? Is $100 a fair premium for this policy? MATH 110 Sec 13-4 Lecture: Expected Value Add up all the expected payouts to get: $40 + $12 + $6 + $32 = $90. Let’s look more closely at the calculation giving a $90 expected payout.

19. The value of items along with the probabilities that they will be stolen over the next year are shown. What can the insurance company expect to pay in claims on your policy? Is $100 a fair premium for this policy? MATH 110 Sec 13-4 Lecture: Expected Value This leads us to a general definition. DEFINITION If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n

20. The value of items along with the probabilities that they will be stolen over the next year are shown. What can the insurance company expect to pay in claims on your policy? Is $100 a fair premium for this policy? MATH 110 Sec 13-4 Lecture: Expected Value This leads us to a general definition. DEFINITION If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n

21. The value of items along with the probabilities that they will be stolen over the next year are shown. What can the insurance company expect to pay in claims on your policy? Is $100 a fair premium for this policy? MATH 110 Sec 13-4 Lecture: Expected Value This leads us to a general definition. DEFINITION If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n Let’s begin with a simple example.

22. MATH 110 Sec 13-4 Lecture: Expected Value To the right are the probabilities and values associated with five outcomes of an experiment. Calculate the expected value for the experiment. OutcomeProbabilityValueProb x Value A0.15 B0.25 C -2 D0.2-2 E0.39

23. MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n To the right are the probabilities and values associated with five outcomes of an experiment. Calculate the expected value for the experiment. OutcomeProbabilityValueProb x Value A0.15 B0.25 C -2 D0.2-2 E0.39

24. MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n To the right are the probabilities and values associated with five outcomes of an experiment. Calculate the expected value for the experiment. OutcomeProbabilityValueProb x Value A0.15 B0.25 C -2 D0.2-2 E0.39 If the information is already tabled, it is simpler to just add a column for the required products.

25. MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n To the right are the probabilities and values associated with five outcomes of an experiment. Calculate the expected value for the experiment. OutcomeProbabilityValueProb x Value A0.15 B0.25 C -2 D0.2-2 E0.39 If the information is already tabled, it is simpler to just add a column for the required products.

26. MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n To the right are the probabilities and values associated with five outcomes of an experiment. Calculate the expected value for the experiment. OutcomeProbabilityValue A0.15 B0.25 C -2 D0.2-2 E0.39 If the information is already tabled, it is simpler to just add a column for the required products.

27. MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n To the right are the probabilities and values associated with five outcomes of an experiment. Calculate the expected value for the experiment. OutcomeProbabilityValue A0.15 B0.25 C -2 D0.2-2 E0.39 If the information is already tabled, it is simpler to just add a column for the required products.

28. MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n To the right are the probabilities and values associated with five outcomes of an experiment. Calculate the expected value for the experiment. OutcomeProbabilityValue A0.15 0.5 B0.25 C -2 D0.2-2 E0.39 If the information is already tabled, it is simpler to just add a column for the required products.

29. MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n To the right are the probabilities and values associated with five outcomes of an experiment. Calculate the expected value for the experiment. OutcomeProbabilityValue A0.15 0.5 B0.25 C -2 D0.2-2 E0.39 If the information is already tabled, it is simpler to just add a column for the required products.

30. MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n To the right are the probabilities and values associated with five outcomes of an experiment. Calculate the expected value for the experiment. OutcomeProbabilityValue A0.15 0.5 B0.25 1.0 C0.2-2 D0.2-2 E0.39 If the information is already tabled, it is simpler to just add a column for the required products.

31. MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n To the right are the probabilities and values associated with five outcomes of an experiment. Calculate the expected value for the experiment. OutcomeProbabilityValue A0.15 0.5 B0.25 1.0 C0.2-2 D0.2-2 E0.39 If the information is already tabled, it is simpler to just add a column for the required products.

32. MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n To the right are the probabilities and values associated with five outcomes of an experiment. Calculate the expected value for the experiment. OutcomeProbabilityValue A0.15 0.5 B0.25 1.0 C0.2-2-0.4 D0.2-2 E0.39 If the information is already tabled, it is simpler to just add a column for the required products.

33. MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n To the right are the probabilities and values associated with five outcomes of an experiment. Calculate the expected value for the experiment. OutcomeProbabilityValue A0.15 0.5 B0.25 1.0 C0.2-2-0.4 D0.2-2 E0.39 If the information is already tabled, it is simpler to just add a column for the required products.

34. MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n To the right are the probabilities and values associated with five outcomes of an experiment. Calculate the expected value for the experiment. OutcomeProbabilityValue A0.15 0.5 B0.25 1.0 C0.2-2-0.4 D0.2-2-0.4 E0.39 If the information is already tabled, it is simpler to just add a column for the required products.

35. MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n To the right are the probabilities and values associated with five outcomes of an experiment. Calculate the expected value for the experiment. OutcomeProbabilityValue A0.15 0.5 B0.25 1.0 C0.2-2-0.4 D0.2-2-0.4 E0.39 If the information is already tabled, it is simpler to just add a column for the required products.

36. MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n To the right are the probabilities and values associated with five outcomes of an experiment. Calculate the expected value for the experiment. OutcomeProbabilityValue A0.15 0.5 B0.25 1.0 C0.2-2-0.4 D0.2-2-0.4 E0.39 2.7 If the information is already tabled, it is simpler to just add a column for the required products.

37. MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n To the right are the probabilities and values associated with five outcomes of an experiment. Calculate the expected value for the experiment. OutcomeProbabilityValue A0.15 0.5 B0.25 1.0 C0.2-2-0.4 D0.2-2-0.4 E0.39 2.7 If the information is already tabled, it is simpler to just add a column for the required products.

38. MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n To the right are the probabilities and values associated with five outcomes of an experiment. Calculate the expected value for the experiment. OutcomeProbabilityValue A0.15 0.5 B0.25 1.0 C0.2-2-0.4 D0.2-2-0.4 E0.39 2.7 The sum of these products is the expected value for the experiment.

39. MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n To the right are the probabilities and values associated with five outcomes of an experiment. Calculate the expected value for the experiment. OutcomeProbabilityValue A0.15 0.5 B0.25 1.0 C0.2-2-0.4 D0.2-2-0.4 E0.39 2.7 The sum of these products is the expected value for the experiment. 0.5 + 1.0 – 0.4 – 0.4 + 2.7 = 3.4

40. MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n To the right are the probabilities and values associated with five outcomes of an experiment. Calculate the expected value for the experiment. OutcomeProbabilityValue A0.15 0.5 B0.25 1.0 C0.2-2-0.4 D0.2-2-0.4 E0.39 2.7 The sum of these products is the expected value for the experiment. 0.5 + 1.0 – 0.4 – 0.4 + 2.7 = 3.4 3.4

41. MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n To the right are the probabilities and values associated with five outcomes of an experiment. Calculate the expected value for the experiment. OutcomeProbabilityValue A0.15 0.5 B0.25 1.0 C0.2-2-0.4 D0.2-2-0.4 E0.39 2.7 The expected value for this experiment is 3.4

42. MATH 110 Sec 13-4 Lecture: Expected Value To the right are the probabilities and values associated with five outcomes of an experiment. Calculate the expected value for the experiment. OutcomeProbabilityValue A0.15 0.5 B0.25 1.0 C0.2-2-0.4 D0.2-2-0.4 E0.39 2.7 The expected value for this experiment is 3.4 Let’s look at another example of finding the expected value.

43. MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n Example: How many tails can we expect when we flip 4 fair coins?

44. MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n H T H T H T H T H T H T H T Example: How many tails can we expect when we flip 4 fair coins? As a visual aid, let’s use this tree diagram that we’ve constructed before.

45. There are C(4,0)=1 way to get 0 tails There’s C(4,1)=4 ways to get 1 tail There’s C(4,2)=6 ways to get 2 tails There’s C(4,3)=4 ways to get 3 tails There’s C(4,4)=1 way to get 4 tails MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n H T H T H T H T H T H T H T Example: How many tails can we expect when we flip 4 fair coins? { HHHH, HHHT, HHTH, HTHH, THHH, HHTT, HTHT, HTTH, THHT, THTH, TTHH, HTTT, THTT, TTHT, TTTH, TTTT }

46. There are C(4,0)=1 way to get 0 tails There are C(4,1)=4 ways to get 1 tail There’s C(4,2)=6 ways to get 2 tails There’s C(4,3)=4 ways to get 3 tails There’s C(4,4)=1 way to get 4 tails MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n H T H T H T H T H T H T H T Example: How many tails can we expect when we flip 4 fair coins?

47. { HHHH, HHHT, HHTH, HTHH, THHH, HHTT, HTHT, HTTH, THHT, THTH, TTHH, HTTT, THTT, TTHT, TTTH, TTTT } There are C(4,0)=1 way to get 0 tails There are C(4,1)=4 ways to get 1 tail There are C(4,2)=6 ways to get 2 tails There’s C(4,3)=4 ways to get 3 tails There’s C(4,4)=1 way to get 4 tails MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n H T H T H T H T H T H T H T Example: How many tails can we expect when we flip 4 fair coins?

48. { HHHH, HHHT, HHTH, HTHH, THHH, HHTT, HTHT, HTTH, THHT, THTH, TTHH, HTTT, THTT, TTHT, TTTH, TTTT } There are C(4,0)=1 way to get 0 tails There are C(4,1)=4 ways to get 1 tail There are C(4,2)=6 ways to get 2 tails There are C(4,3)=4 ways to get 3 tails There’s C(4,4)=1 way to get 4 tails MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n H T H T H T H T H T H T H T Example: How many tails can we expect when we flip 4 fair coins?

49. { HHHH, HHHT, HHTH, HTHH, THHH, HHTT, HTHT, HTTH, THHT, THTH, TTHH, HTTT, THTT, TTHT, TTTH, TTTT } There are C(4,0)=1 way to get 0 tails There are C(4,1)=4 ways to get 1 tail There are C(4,2)=6 ways to get 2 tails There are C(4,3)=4 ways to get 3 tails There are C(4,4)=1 way to get 4 tails MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n H T H T H T H T H T H T H T Example: How many tails can we expect when we flip 4 fair coins?

50. MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n H T H T H T H T H T H T H T Example: How many tails can we expect when we flip 4 fair coins? There are C(4,0)=1 way to get 0 tails There are C(4,1)=4 ways to get 1 tail There are C(4,2)=6 ways to get 2 tails There are C(4,3)=4 ways to get 3 tails There are C(4,4)=1 way to get 4 tails { HHHH, HHHT, HHTH, HTHH, THHH, HHTT, HTHT, HTTH, THHT, THTH, TTHH, HTTT, THTT, TTHT, TTTH, TTTT }

51. MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n H T H T H T H T H T H T H T Example: How many tails can we expect when we flip 4 fair coins? There are C(4,0)=1 way to get 0 tails There are C(4,1)=4 ways to get 1 tail There are C(4,2)=6 ways to get 2 tails There are C(4,3)=4 ways to get 3 tails There are C(4,4)=1 way to get 4 tails In all, there are 16 different ways. { HHHH, HHHT, HHTH, HTHH, THHH, HHTT, HTHT, HTTH, THHT, THTH, TTHH, HTTT, THTT, TTHT, TTTH, TTTT }

52. MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n H T H T H T H T H T H T H T Example: How many tails can we expect when we flip 4 fair coins? There are C(4,0)=1 way to get 0 tails There are C(4,1)=4 ways to get 1 tail There are C(4,2)=6 ways to get 2 tails There are C(4,3)=4 ways to get 3 tails There are C(4,4)=1 way to get 4 tails In all, there are 16 different ways. { HHHH, HHHT, HHTH, HTHH, THHH, HHTT, HTHT, HTTH, THHT, THTH, TTHH, HTTT, THTT, TTHT, TTTH, TTTT }

53. MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n H T H T H T H T H T H T H T { HHHH, HTTT, THTT, TTHT, TTTH, HHTT, HTHT, HTTH, THHT, THTH, TTHH,, HHHT, HHTH, HTHH, THHH, TTTT } Example: How many tails can we expect when we flip 4 fair coins? Tabling these values, we can easily compute the probability of each number of tails. There are C(4,0)=1 way to get 0 tails There are C(4,1)=4 ways to get 1 tail There are C(4,2)=6 ways to get 2 tails There are C(4,3)=4 ways to get 3 tails There are C(4,4)=1 way to get 4 tails # of TAILSProbability# TAILS x Probability 0 1 2 3 4 In all, there are 16 different ways.

54. MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n H T H T H T H T H T H T H T { HHHH, HTTT, THTT, TTHT, TTTH, HHTT, HTHT, HTTH, THHT, THTH, TTHH,, HHHT, HHTH, HTHH, THHH, TTTT } Example: How many tails can we expect when we flip 4 fair coins? Tabling these values, we can easily compute the probability of each number of tails. There are C(4,0)=1 way to get 0 tails There are C(4,1)=4 ways to get 1 tail There are C(4,2)=6 ways to get 2 tails There are C(4,3)=4 ways to get 3 tails There are C(4,4)=1 way to get 4 tails # of TAILSProbability# TAILS x Probability 0 1 2 3 4 In all, there are 16 different ways.

55. There are C(4,0)=1 way to get 0 tails There are C(4,1)=4 ways to get 1 tail There are C(4,2)=6 ways to get 2 tails There are C(4,3)=4 ways to get 3 tails There are C(4,4)=1 way to get 4 tails MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n H T H T H T H T H T H T H T { HHHH, HTTT, THTT, TTHT, TTTH, HHTT, HTHT, HTTH, THHT, THTH, TTHH,, HHHT, HHTH, HTHH, THHH, TTTT } Example: How many tails can we expect when we flip 4 fair coins? Tabling these values, we can easily compute the probability of each number of tails. # of TAILSProbability# TAILS x Probability 0 1 2 3 4 In all, there are 16 different ways.

56. There are C(4,0)=1 way to get 0 tails There are C(4,1)=4 ways to get 1 tail There are C(4,2)=6 ways to get 2 tails There are C(4,3)=4 ways to get 3 tails There are C(4,4)=1 way to get 4 tails MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n H T H T H T H T H T H T H T { HHHH, HTTT, THTT, TTHT, TTTH, HHTT, HTHT, HTTH, THHT, THTH, TTHH,, HHHT, HHTH, HTHH, THHH, TTTT } Example: How many tails can we expect when we flip 4 fair coins? Tabling these values, we can easily compute the probability of each number of tails. # of TAILSProbability# TAILS x Probability 0 1 2 3 4 In all, there are 16 different ways.

57. There are C(4,0)=1 way to get 0 tails There are C(4,1)=4 ways to get 1 tail There are C(4,2)=6 ways to get 2 tails There are C(4,3)=4 ways to get 3 tails There are C(4,4)=1 way to get 4 tails MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n H T H T H T H T H T H T H T { HHHH, HTTT, THTT, TTHT, TTTH, HHTT, HTHT, HTTH, THHT, THTH, TTHH,, HHHT, HHTH, HTHH, THHH, TTTT } Example: How many tails can we expect when we flip 4 fair coins? Tabling these values, we can easily compute the probability of each number of tails. # of TAILSProbability# TAILS x Probability 0 1 2 3 4 In all, there are 16 different ways.

58. There are C(4,0)=1 way to get 0 tails There are C(4,1)=4 ways to get 1 tail There are C(4,2)=6 ways to get 2 tails There are C(4,3)=4 ways to get 3 tails There are C(4,4)=1 way to get 4 tails MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n H T H T H T H T H T H T H T { HHHH, HTTT, THTT, TTHT, TTTH, HHTT, HTHT, HTTH, THHT, THTH, TTHH,, HHHT, HHTH, HTHH, THHH, TTTT } Example: How many tails can we expect when we flip 4 fair coins? Tabling these values, we can easily compute the probability of each number of tails. # of TAILSProbability# TAILS x Probability 0 1 2 3 4 In all, there are 16 different ways.

59. There are C(4,0)=1 way to get 0 tails There are C(4,1)=4 ways to get 1 tail There are C(4,2)=6 ways to get 2 tails There are C(4,3)=4 ways to get 3 tails There are C(4,4)=1 way to get 4 tails MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n H T H T H T H T H T H T H T { HHHH, HTTT, THTT, TTHT, TTTH, HHTT, HTHT, HTTH, THHT, THTH, TTHH,, HHHT, HHTH, HTHH, THHH, TTTT } Example: How many tails can we expect when we flip 4 fair coins? Tabling these values, we can easily compute the probability of each number of tails. # of TAILSProbability# TAILS x Probability 0 1 2 3 4 In all, there are 16 different ways.

60. There are C(4,0)=1 way to get 0 tails There are C(4,1)=4 ways to get 1 tail There are C(4,2)=6 ways to get 2 tails There are C(4,3)=4 ways to get 3 tails There are C(4,4)=1 way to get 4 tails MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n H T H T H T H T H T H T H T { HHHH, HTTT, THTT, TTHT, TTTH, HHTT, HTHT, HTTH, THHT, THTH, TTHH,, HHHT, HHTH, HTHH, THHH, TTTT } Example: How many tails can we expect when we flip 4 fair coins? Tabling these values, we can easily compute the probability of each number of tails. # of TAILSProbability# TAILS x Probability 0 1 2 3 4 In all, there are 16 different ways.

61. There are C(4,0)=1 way to get 0 tails There are C(4,1)=4 ways to get 1 tail There are C(4,2)=6 ways to get 2 tails There are C(4,3)=4 ways to get 3 tails There are C(4,4)=1 way to get 4 tails MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n H T H T H T H T H T H T H T { HHHH, HTTT, THTT, TTHT, TTTH, HHTT, HTHT, HTTH, THHT, THTH, TTHH,, HHHT, HHTH, HTHH, THHH, TTTT } Example: How many tails can we expect when we flip 4 fair coins? Tabling these values, we can easily compute the probability of each number of tails. # of TAILSProbability# TAILS x Probability 0 1 2 3 4 In all, there are 16 different ways.

62. There are C(4,0)=1 way to get 0 tails There are C(4,1)=4 ways to get 1 tail There are C(4,2)=6 ways to get 2 tails There are C(4,3)=4 ways to get 3 tails There are C(4,4)=1 way to get 4 tails MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n H T H T H T H T H T H T H T { HHHH, HTTT, THTT, TTHT, TTTH, HHTT, HTHT, HTTH, THHT, THTH, TTHH,, HHHT, HHTH, HTHH, THHH, TTTT } Example: How many tails can we expect when we flip 4 fair coins? Tabling these values, we can easily compute the probability of each number of tails. # of TAILSProbability# TAILS x Probability 0 1 2 3 4 In all, there are 16 different ways.

63. There are C(4,0)=1 way to get 0 tails There are C(4,1)=4 ways to get 1 tail There are C(4,2)=6 ways to get 2 tails There are C(4,3)=4 ways to get 3 tails There are C(4,4)=1 way to get 4 tails MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n H T H T H T H T H T H T H T { HHHH, HTTT, THTT, TTHT, TTTH, HHTT, HTHT, HTTH, THHT, THTH, TTHH,, HHHT, HHTH, HTHH, THHH, TTTT } Example: How many tails can we expect when we flip 4 fair coins? Tabling these values, we can easily compute the probability of each number of tails. # of TAILSProbability# TAILS x Probability 0 1 2 3 4 In all, there are 16 different ways.

64. There are C(4,0)=1 way to get 0 tails There are C(4,1)=4 ways to get 1 tail There are C(4,2)=6 ways to get 2 tails There are C(4,3)=4 ways to get 3 tails There are C(4,4)=1 way to get 4 tails MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n H T H T H T H T H T H T H T { HHHH, HTTT, THTT, TTHT, TTTH, HHTT, HTHT, HTTH, THHT, THTH, TTHH,, HHHT, HHTH, HTHH, THHH, TTTT } Example: How many tails can we expect when we flip 4 fair coins? Tabling these values, we can easily compute the probability of each number of tails. # of TAILSProbability# TAILS x Probability 0 1 2 3 4 In all, there are 16 different ways.

65. There are C(4,0)=1 way to get 0 tails There are C(4,1)=4 ways to get 1 tail There are C(4,2)=6 ways to get 2 tails There are C(4,3)=4 ways to get 3 tails There are C(4,4)=1 way to get 4 tails MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n H T H T H T H T H T H T H T { HHHH, HTTT, THTT, TTHT, TTTH, HHTT, HTHT, HTTH, THHT, THTH, TTHH,, HHHT, HHTH, HTHH, THHH, TTTT } Example: How many tails can we expect when we flip 4 fair coins? Tabling these values, we can easily compute the probability of each number of tails. # of TAILSProbability# TAILS x Probability 0 1 2 3 4 In all, there are 16 different ways.

66. There are C(4,0)=1 way to get 0 tails There are C(4,1)=4 ways to get 1 tail There are C(4,2)=6 ways to get 2 tails There are C(4,3)=4 ways to get 3 tails There are C(4,4)=1 way to get 4 tails MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n H T H T H T H T H T H T H T { HHHH, HTTT, THTT, TTHT, TTTH, HHTT, HTHT, HTTH, THHT, THTH, TTHH,, HHHT, HHTH, HTHH, THHH, TTTT } Example: How many tails can we expect when we flip 4 fair coins? Tabling these values, we can easily compute the probability of each number of tails. # of TAILSProbability# TAILS x Probability 0 1 2 3 4 In all, there are 16 different ways.

67. There are C(4,0)=1 way to get 0 tails There are C(4,1)=4 ways to get 1 tail There are C(4,2)=6 ways to get 2 tails There are C(4,3)=4 ways to get 3 tails There are C(4,4)=1 way to get 4 tails MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n H T H T H T H T H T H T H T { HHHH, HTTT, THTT, TTHT, TTTH, HHTT, HTHT, HTTH, THHT, THTH, TTHH,, HHHT, HHTH, HTHH, THHH, TTTT } Example: How many tails can we expect when we flip 4 fair coins? Tabling these values, we can easily compute the probability of each number of tails. # of TAILSProbability# TAILS x Probability 0 1 2 3 4 In all, there are 16 different ways.

68. There are C(4,0)=1 way to get 0 tails There are C(4,1)=4 ways to get 1 tail There are C(4,2)=6 ways to get 2 tails There are C(4,3)=4 ways to get 3 tails There are C(4,4)=1 way to get 4 tails MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n H T H T H T H T H T H T H T { HHHH, HTTT, THTT, TTHT, TTTH, HHTT, HTHT, HTTH, THHT, THTH, TTHH,, HHHT, HHTH, HTHH, THHH, TTTT } Example: How many tails can we expect when we flip 4 fair coins? Tabling these values, we can easily compute the probability of each number of tails. # of TAILSProbability# TAILS x Probability 0 1 2 3 4 In all, there are 16 different ways.

69. There are C(4,0)=1 way to get 0 tails There are C(4,1)=4 ways to get 1 tail There are C(4,2)=6 ways to get 2 tails There are C(4,3)=4 ways to get 3 tails There are C(4,4)=1 way to get 4 tails MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n H T H T H T H T H T H T H T { HHHH, HTTT, THTT, TTHT, TTTH, HHTT, HTHT, HTTH, THHT, THTH, TTHH,, HHHT, HHTH, HTHH, THHH, TTTT } Example: How many tails can we expect when we flip 4 fair coins? Tabling these values, we can easily compute the probability of each number of tails. # of TAILSProbability# TAILS x Probability 0 1 2 3 4 In all, there are 16 different ways.

70. There are C(4,0)=1 way to get 0 tails There are C(4,1)=4 ways to get 1 tail There are C(4,2)=6 ways to get 2 tails There are C(4,3)=4 ways to get 3 tails There are C(4,4)=1 way to get 4 tails MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n H T H T H T H T H T H T H T { HHHH, HTTT, THTT, TTHT, TTTH, HHTT, HTHT, HTTH, THHT, THTH, TTHH,, HHHT, HHTH, HTHH, THHH, TTTT } Example: How many tails can we expect when we flip 4 fair coins? Tabling these values, we can easily compute the probability of each number of tails. # of TAILSProbability# TAILS x Probability 0 1 2 3 4 In all, there are 16 different ways.

71. There are C(4,0)=1 way to get 0 tails There are C(4,1)=4 ways to get 1 tail There are C(4,2)=6 ways to get 2 tails There are C(4,3)=4 ways to get 3 tails There are C(4,4)=1 way to get 4 tails MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n H T H T H T H T H T H T H T { HHHH, HTTT, THTT, TTHT, TTTH, HHTT, HTHT, HTTH, THHT, THTH, TTHH,, HHHT, HHTH, HTHH, THHH, TTTT } Example: How many tails can we expect when we flip 4 fair coins? Tabling these values, we can easily compute the probability of each number of tails. # of TAILSProbability# TAILS x Probability 0 1 2 3 4 In all, there are 16 different ways.

72. There are C(4,0)=1 way to get 0 tails There are C(4,1)=4 ways to get 1 tail There are C(4,2)=6 ways to get 2 tails There are C(4,3)=4 ways to get 3 tails There are C(4,4)=1 way to get 4 tails MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n H T H T H T H T H T H T H T { HHHH, HTTT, THTT, TTHT, TTTH, HHTT, HTHT, HTTH, THHT, THTH, TTHH,, HHHT, HHTH, HTHH, THHH, TTTT } Example: How many tails can we expect when we flip 4 fair coins? Tabling these values, we can easily compute the probability of each number of tails. # of TAILSProbability# TAILS x Probability 0 1 2 3 4 In all, there are 16 different ways.

73. There are C(4,0)=1 way to get 0 tails There are C(4,1)=4 ways to get 1 tail There are C(4,2)=6 ways to get 2 tails There are C(4,3)=4 ways to get 3 tails There are C(4,4)=1 way to get 4 tails MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n H T H T H T H T H T H T H T { HHHH, HTTT, THTT, TTHT, TTTH, HHTT, HTHT, HTTH, THHT, THTH, TTHH,, HHHT, HHTH, HTHH, THHH, TTTT } Example: How many tails can we expect when we flip 4 fair coins? Tabling these values, we can easily compute the probability of each number of tails. # of TAILSProbability# TAILS x Probability 0 1 2 3 4 In all, there are 16 different ways.

74. There are C(4,0)=1 way to get 0 tails There are C(4,1)=4 ways to get 1 tail There are C(4,2)=6 ways to get 2 tails There are C(4,3)=4 ways to get 3 tails There are C(4,4)=1 way to get 4 tails MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n H T H T H T H T H T H T H T { HHHH, HTTT, THTT, TTHT, TTTH, HHTT, HTHT, HTTH, THHT, THTH, TTHH,, HHHT, HHTH, HTHH, THHH, TTTT } Example: How many tails can we expect when we flip 4 fair coins? Tabling these values, we can easily compute the probability of each number of tails. # of TAILSProbability# TAILS x Probability 0 1 2 3 4 In all, there are 16 different ways.

75. There are C(4,0)=1 way to get 0 tails There are C(4,1)=4 ways to get 1 tail There are C(4,2)=6 ways to get 2 tails There are C(4,3)=4 ways to get 3 tails There are C(4,4)=1 way to get 4 tails MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n H T H T H T H T H T H T H T { HHHH, HTTT, THTT, TTHT, TTTH, HHTT, HTHT, HTTH, THHT, THTH, TTHH,, HHHT, HHTH, HTHH, THHH, TTTT } Example: How many tails can we expect when we flip 4 fair coins? Tabling these values, we can easily compute the probability of each number of tails. # of TAILSProbability# TAILS x Probability 0 1 2 3 4 In all, there are 16 different ways.

76. There are C(4,0)=1 way to get 0 tails There are C(4,1)=4 ways to get 1 tail There are C(4,2)=6 ways to get 2 tails There are C(4,3)=4 ways to get 3 tails There are C(4,4)=1 way to get 4 tails MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n H T H T H T H T H T H T H T { HHHH, HTTT, THTT, TTHT, TTTH, HHTT, HTHT, HTTH, THHT, THTH, TTHH,, HHHT, HHTH, HTHH, THHH, TTTT } Example: How many tails can we expect when we flip 4 fair coins? Tabling these values, we can easily compute the probability of each number of tails. # of TAILSProbability# TAILS x Probability 0 1 2 3 4 In all, there are 16 different ways.

77. There are C(4,0)=1 way to get 0 tails There are C(4,1)=4 ways to get 1 tail There are C(4,2)=6 ways to get 2 tails There are C(4,3)=4 ways to get 3 tails There are C(4,4)=1 way to get 4 tails MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n H T H T H T H T H T H T H T { HHHH, HTTT, THTT, TTHT, TTTH, HHTT, HTHT, HTTH, THHT, THTH, TTHH,, HHHT, HHTH, HTHH, THHH, TTTT } Example: How many tails can we expect when we flip 4 fair coins? Tabling these values, we can easily compute the probability of each number of tails. # of TAILSProbability# TAILS x Probability 0 1 2 3 4 In all, there are 16 different ways.

78. There are C(4,0)=1 way to get 0 tails There are C(4,1)=4 ways to get 1 tail There are C(4,2)=6 ways to get 2 tails There are C(4,3)=4 ways to get 3 tails There are C(4,4)=1 way to get 4 tails MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n H T H T H T H T H T H T H T { HHHH, HTTT, THTT, TTHT, TTTH, HHTT, HTHT, HTTH, THHT, THTH, TTHH,, HHHT, HHTH, HTHH, THHH, TTTT } Example: How many tails can we expect when we flip 4 fair coins? Tabling these values, we can easily compute the probability of each number of tails. # of TAILSProbability# TAILS x Probability 0 1 2 3 4 In all, there are 16 different ways.

79. There are C(4,0)=1 way to get 0 tails There are C(4,1)=4 ways to get 1 tail There are C(4,2)=6 ways to get 2 tails There are C(4,3)=4 ways to get 3 tails There are C(4,4)=1 way to get 4 tails MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n H T H T H T H T H T H T H T { HHHH, HTTT, THTT, TTHT, TTTH, HHTT, HTHT, HTTH, THHT, THTH, TTHH,, HHHT, HHTH, HTHH, THHH, TTTT } Example: How many tails can we expect when we flip 4 fair coins? Tabling these values, we can easily compute the probability of each number of tails. # of TAILSProbability# TAILS x Probability 0 1 2 3 4 In all, there are 16 different ways.

80. There are C(4,0)=1 way to get 0 tails There are C(4,1)=4 ways to get 1 tail There are C(4,2)=6 ways to get 2 tails There are C(4,3)=4 ways to get 3 tails There are C(4,4)=1 way to get 4 tails MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n H T H T H T H T H T H T H T { HHHH, HTTT, THTT, TTHT, TTTH, HHTT, HTHT, HTTH, THHT, THTH, TTHH,, HHHT, HHTH, HTHH, THHH, TTTT } Example: How many tails can we expect when we flip 4 fair coins? Tabling these values, we can easily compute the probability of each number of tails. # of TAILSProbability# TAILS x Probability 0 1 2 3 4 In all, there are 16 different ways.

81. There are C(4,0)=1 way to get 0 tails There are C(4,1)=4 ways to get 1 tail There are C(4,2)=6 ways to get 2 tails There are C(4,3)=4 ways to get 3 tails There are C(4,4)=1 way to get 4 tails MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n H T H T H T H T H T H T H T { HHHH, HTTT, THTT, TTHT, TTTH, HHTT, HTHT, HTTH, THHT, THTH, TTHH,, HHHT, HHTH, HTHH, THHH, TTTT } Example: How many tails can we expect when we flip 4 fair coins? Tabling these values, we can easily compute the probability of each number of tails. # of TAILSProbability# TAILS x Probability 0 1 2 3 4 In all, there are 16 different ways.

82. There are C(4,0)=1 way to get 0 tails There are C(4,1)=4 ways to get 1 tail There are C(4,2)=6 ways to get 2 tails There are C(4,3)=4 ways to get 3 tails There are C(4,4)=1 way to get 4 tails MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n H T H T H T H T H T H T H T { HHHH, HTTT, THTT, TTHT, TTTH, HHTT, HTHT, HTTH, THHT, THTH, TTHH,, HHHT, HHTH, HTHH, THHH, TTTT } Example: How many tails can we expect when we flip 4 fair coins? Tabling these values, we can easily compute the probability of each number of tails. # of TAILSProbability# TAILS x Probability 0 1 2 3 4 In all, there are 16 different ways.

83. There are C(4,0)=1 way to get 0 tails There are C(4,1)=4 ways to get 1 tail There are C(4,2)=6 ways to get 2 tails There are C(4,3)=4 ways to get 3 tails There are C(4,4)=1 way to get 4 tails MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n H T H T H T H T H T H T H T { HHHH, HTTT, THTT, TTHT, TTTH, HHTT, HTHT, HTTH, THHT, THTH, TTHH,, HHHT, HHTH, HTHH, THHH, TTTT } Example: How many tails can we expect when we flip 4 fair coins? Tabling these values, we can easily compute the probability of each number of tails. # of TAILSProbability# TAILS x Probability 0 1 2 3 4 In all, there are 16 different ways.

84. There are C(4,0)=1 way to get 0 tails There are C(4,1)=4 ways to get 1 tail There are C(4,2)=6 ways to get 2 tails There are C(4,3)=4 ways to get 3 tails There are C(4,4)=1 way to get 4 tails MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n H T H T H T H T H T H T H T { HHHH, HTTT, THTT, TTHT, TTTH, HHTT, HTHT, HTTH, THHT, THTH, TTHH,, HHHT, HHTH, HTHH, THHH, TTTT } Example: How many tails can we expect when we flip 4 fair coins? Tabling these values, we can easily compute the probability of each number of tails. # of TAILSProbability# TAILS x Probability 0 1 2 3 4 In all, there are 16 different ways.

85. There are C(4,0)=1 way to get 0 tails There are C(4,1)=4 ways to get 1 tail There are C(4,2)=6 ways to get 2 tails There are C(4,3)=4 ways to get 3 tails There are C(4,4)=1 way to get 4 tails MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n H T H T H T H T H T H T H T { HHHH, HTTT, THTT, TTHT, TTTH, HHTT, HTHT, HTTH, THHT, THTH, TTHH,, HHHT, HHTH, HTHH, THHH, TTTT } Example: How many tails can we expect when we flip 4 fair coins? Tabling these values, we can easily compute the probability of each number of tails. # of TAILSProbability# TAILS x Probability 0 1 2 3 4 In all, there are 16 different ways.

86. There are C(4,0)=1 way to get 0 tails There are C(4,1)=4 ways to get 1 tail There are C(4,2)=6 ways to get 2 tails There are C(4,3)=4 ways to get 3 tails There are C(4,4)=1 way to get 4 tails MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n { HHHH, HTTT, THTT, TTHT, TTTH, HHTT, HTHT, HTTH, THHT, THTH, TTHH,, HHHT, HHTH, HTHH, THHH, TTTT } Example: How many tails can we expect when we flip 4 fair coins? Tabling these values, we can easily compute the probability of each number of tails. # of TAILSProbability# TAILS x Probability 0 1 2 3 4 In all, there are 16 different ways.

87. There are C(4,0)=1 way to get 0 tails There are C(4,1)=4 ways to get 1 tail There are C(4,2)=6 ways to get 2 tails There are C(4,3)=4 ways to get 3 tails There are C(4,4)=1 way to get 4 tails MATH 110 Sec 13-4 Lecture: Expected Value If an experiment has numerical outcomes x 1, x 2, x 3,..., x n With corresponding probabilities p 1, p 2, p 3,..., p n, then the expected value of the experiment is given by x 1 p 1 + x 2 p 2 + x 3 p 3 +... + x n p n { HHHH, HTTT, THTT, TTHT, TTTH, HHTT, HTHT, HTTH, THHT, THTH, TTHH,, HHHT, HHTH, HTHH, THHH, TTTT } Example: How many tails can we expect when we flip 4 fair coins? Tabling these values, we can easily compute the probability of each number of tails. # of TAILSProbability# TAILS x Probability 0 1 2 3 4 In all, there are 16 different ways. So, the expected number of tails is 2.

88. MATH 110 Sec 13-4 Lecture: Expected Value Suppose you bet $1 on a single number of a 38 number roulette wheel. If you win, you get $35 (and also keep your $1 bet)…otherwise you lose $1. What is the expected value of this bet (to the nearest cent)?

89. MATH 110 Sec 13-4 Lecture: Expected Value Suppose you bet $1 on a single number of a 38 number roulette wheel. If you win, you get $35 (and also keep your $1 bet)…otherwise you lose $1. What is the expected value of this bet (to the nearest cent)? If you are betting on a single number (out of 38), then you have a probability of 1 / 38 of winning $35.

90. MATH 110 Sec 13-4 Lecture: Expected Value Suppose you bet $1 on a single number of a 38 number roulette wheel. If you win, you get $35 (and also keep your $1 bet)…otherwise you lose $1. What is the expected value of this bet (to the nearest cent)? If you are betting on a single number (out of 38), then you have a probability of 1 / 38 of winning $35. Amt won or lostProbability Winnings x Probability $35 1 / 38 ─ $1 37 / 38

91. MATH 110 Sec 13-4 Lecture: Expected Value Suppose you bet $1 on a single number of a 38 number roulette wheel. If you win, you get $35 (and also keep your $1 bet)…otherwise you lose $1. What is the expected value of this bet (to the nearest cent)? If you are betting on a single number (out of 38), then you have a probability of 1 / 38 of winning $35. Amt won or lostProbability Winnings x Probability $35 1 / 38 ─ $1 37 / 38

92. MATH 110 Sec 13-4 Lecture: Expected Value Suppose you bet $1 on a single number of a 38 number roulette wheel. If you win, you get $35 (and also keep your $1 bet)…otherwise you lose $1. What is the expected value of this bet (to the nearest cent)? Amt won or lostProbability Winnings x Probability $35 1 / 38 ─ $1 37 / 38 If you are betting on a single number (out of 38), then you have a probability of 1 / 38 of winning $35. Of course, that means that you have a 37 / 38 chance of losing your $1 bet.

93. MATH 110 Sec 13-4 Lecture: Expected Value Suppose you bet $1 on a single number of a 38 number roulette wheel. If you win, you get $35 (and also keep your $1 bet)…otherwise you lose $1. What is the expected value of this bet (to the nearest cent)? Amt won or lostProbability Winnings x Probability $35 1 / 38 ─ $1 37 / 38 If you are betting on a single number (out of 38), then you have a probability of 1 / 38 of winning $35. Of course, that means that you have a 37 / 38 chance of losing your $1 bet.

94. MATH 110 Sec 13-4 Lecture: Expected Value Suppose you bet $1 on a single number of a 38 number roulette wheel. If you win, you get $35 (and also keep your $1 bet)…otherwise you lose $1. What is the expected value of this bet (to the nearest cent)? Amt won or lostProbability Winnings x Probability $35 1 / 38 ─ $1 37 / 38 Amt won or lostProbability Winnings x Probability $35 1 / 38 ─ $1 37 / 38 If you are betting on a single number (out of 38), then you have a probability of 1 / 38 of winning $35. Of course, that means that you have a 37 / 38 chance of losing your $1 bet.

95. MATH 110 Sec 13-4 Lecture: Expected Value Suppose you bet $1 on a single number of a 38 number roulette wheel. If you win, you get $35 (and also keep your $1 bet)…otherwise you lose $1. What is the expected value of this bet (to the nearest cent)? Amt won or lostProbability Winnings x Probability $35 1 / 38 ─ $1 37 / 38 Amt won or lostProbability Winnings x Probability $35 1 / 38 ─ $1 37 / 38 If you are betting on a single number (out of 38), then you have a probability of 1 / 38 of winning $35. Of course, that means that you have a 37 / 38 chance of losing your $1 bet.

96. MATH 110 Sec 13-4 Lecture: Expected Value Suppose you bet $1 on a single number of a 38 number roulette wheel. If you win, you get $35 (and also keep your $1 bet)…otherwise you lose $1. What is the expected value of this bet (to the nearest cent)? Of course, that means that you have a 37 / 38 chance of losing your $1 bet. Amt won or lostProbability Winnings x Probability $35 1 / 38 ─ $1 37 / 38 If you are betting on a single number (out of 38), then you have a probability of 1 / 38 of winning $35.

97. MATH 110 Sec 13-4 Lecture: Expected Value Suppose you bet $1 on a single number of a 38 number roulette wheel. If you win, you get $35 (and also keep your $1 bet)…otherwise you lose $1. What is the expected value of this bet (to the nearest cent)? Of course, that means that you have a 37 / 38 chance of losing your $1 bet. Amt won or lostProbability Winnings x Probability $35 1 / 38 ─ $1 37 / 38 If you are betting on a single number (out of 38), then you have a probability of 1 / 38 of winning $35.

98. MATH 110 Sec 13-4 Lecture: Expected Value Suppose you bet $1 on a single number of a 38 number roulette wheel. If you win, you get $35 (and also keep your $1 bet)…otherwise you lose $1. What is the expected value of this bet (to the nearest cent)? Of course, that means that you have a 37 / 38 chance of losing your $1 bet. Amt won or lostProbability Winnings x Probability $35 1 / 38 ─ $1 37 / 38 If you are betting on a single number (out of 38), then you have a probability of 1 / 38 of winning $35.

99. MATH 110 Sec 13-4 Lecture: Expected Value Suppose you bet $1 on a single number of a 38 number roulette wheel. If you win, you get $35 (and also keep your $1 bet)…otherwise you lose $1. What is the expected value of this bet (to the nearest cent)? Of course, that means that you have a 37 / 38 chance of losing your $1 bet. Amt won or lostProbability Winnings x Probability $35 1 / 38 ─ $1 37 / 38 If you are betting on a single number (out of 38), then you have a probability of 1 / 38 of winning $35. Be careful! If you want final answer to be accurate to the nearest cent, you should ALWAYS use more decimal places than that for all intermediate results.

100. MATH 110 Sec 13-4 Lecture: Expected Value Suppose you bet $1 on a single number of a 38 number roulette wheel. If you win, you get $35 (and also keep your $1 bet)…otherwise you lose $1. What is the expected value of this bet (to the nearest cent)? Of course, that means that you have a 37 / 38 chance of losing your $1 bet. Amt won or lostProbability Winnings x Probability $35 1 / 38 ─ $1 37 / 38 If you are betting on a single number (out of 38), then you have a probability of 1 / 38 of winning $35. Be careful! If you want final answer to be accurate to the nearest cent, you should ALWAYS use more decimal places than that for all intermediate results.

101. MATH 110 Sec 13-4 Lecture: Expected Value Suppose you bet $1 on a single number of a 38 number roulette wheel. If you win, you get $35 (and also keep your $1 bet)…otherwise you lose $1. What is the expected value of this bet (to the nearest cent)? Of course, that means that you have a 37 / 38 chance of losing your $1 bet. Amt won or lostProbability Winnings x Probability $35 1 / 38 ─ $1 37 / 38 If you are betting on a single number (out of 38), then you have a probability of 1 / 38 of winning $35. Be careful! If you want final answer to be accurate to the nearest cent, you should ALWAYS use more decimal places than that for all intermediate results.

102. MATH 110 Sec 13-4 Lecture: Expected Value Suppose you bet $1 on a single number of a 38 number roulette wheel. If you win, you get $35 (and also keep your $1 bet)…otherwise you lose $1. What is the expected value of this bet (to the nearest cent)? Of course, that means that you have a 37 / 38 chance of losing your $1 bet. Amt won or lostProbability Winnings x Probability $35 1 / 38 ─ $1 37 / 38 If you are betting on a single number (out of 38), then you have a probability of 1 / 38 of winning $35.

103. MATH 110 Sec 13-4 Lecture: Expected Value Suppose you bet $1 on a single number of a 38 number roulette wheel. If you win, you get $35 (and also keep your $1 bet)…otherwise you lose $1. What is the expected value of this bet (to the nearest cent)? Of course, that means that you have a 37 / 38 chance of losing your $1 bet. Amt won or lostProbability Winnings x Probability $35 1 / 38 ─ $1 37 / 38 If you are betting on a single number (out of 38), then you have a probability of 1 / 38 of winning $35.

104. MATH 110 Sec 13-4 Lecture: Expected Value Suppose you bet $1 on a single number of a 38 number roulette wheel. If you win, you get $35 (and also keep your $1 bet)…otherwise you lose $1. What is the expected value of this bet (to the nearest cent)? Of course, that means that you have a 37 / 38 chance of losing your $1 bet. Amt won or lostProbability Winnings x Probability $35 1 / 38 ─ $1 37 / 38 If you are betting on a single number (out of 38), then you have a probability of 1 / 38 of winning $35. (The negative sign means that this is an expected LOSS.)

105. MATH 110 Sec 13-4 Lecture: Expected Value Suppose you bet $1 on a single number of a 38 number roulette wheel. If you win, you get $35 (and also keep your $1 bet)…otherwise you lose $1. What is the expected value of this bet (to the nearest cent)? Of course, that means that you have a 37 / 38 chance of losing your $1 bet. Amt won or lostProbability Winnings x Probability $35 1 / 38 ─ $1 37 / 38 If you are betting on a single number (out of 38), then you have a probability of 1 / 38 of winning $35. (The negative sign means that this is an expected LOSS.) Another way to look at it is that if you make this bet often, on average, you can expect to lose about 5 cents for every dollar that you bet.

106. MATH 110 Sec 13-4 Lecture: Expected Value Assume that it costs $2 to play a lottery. The player chooses a three-digit number between 000 and 999, inclusive, and if the number is selected, then the player wins $500 ($498 considering the $2 cost to play). What is the expected value of this game (to the nearest cent)?

107. MATH 110 Sec 13-4 Lecture: Expected Value If you are betting on a single 3-digit number (there are 1000 of them), then you have a probability of 1 / 1000 of winning $498. Assume that it costs $2 to play a lottery. The player chooses a three-digit number between 000 and 999, inclusive, and if the number is selected, then the player wins $500 ($498 considering the $2 cost to play). What is the expected value of this game (to the nearest cent)?

108. MATH 110 Sec 13-4 Lecture: Expected Value Amt won or lostProbability Winnings x Probability $498 1 / 1000 ─ $2 998 / 1000 Assume that it costs $2 to play a lottery. The player chooses a three-digit number between 000 and 999, inclusive, and if the number is selected, then the player wins $500 ($498 considering the $2 cost to play). What is the expected value of this game (to the nearest cent)? If you are betting on a single 3-digit number (there are 1000 of them), then you have a probability of 1 / 1000 of winning $498.

109. MATH 110 Sec 13-4 Lecture: Expected Value Assume that it costs $2 to play a lottery. The player chooses a three-digit number between 000 and 999, inclusive, and if the number is selected, then the player wins $500 ($498 considering the $2 cost to play). What is the expected value of this game (to the nearest cent)? If you are betting on a single 3-digit number (there are 1000 of them), then you have a probability of 1 / 1000 of winning $498. Amt won or lostProbability Winnings x Probability $498 1 / 1000 ─ $2 998 / 1000

110. MATH 110 Sec 13-4 Lecture: Expected Value Assume that it costs $2 to play a lottery. The player chooses a three-digit number between 000 and 999, inclusive, and if the number is selected, then the player wins $500 ($498 considering the $2 cost to play). What is the expected value of this game (to the nearest cent)? If you are betting on a single 3-digit number (there are 1000 of them), then you have a probability of 1 / 1000 of winning $498. Amt won or lostProbability Winnings x Probability $498 1 / 1000 ─ $2 998 / 1000

111. MATH 110 Sec 13-4 Lecture: Expected Value And that means that you have a 999 / 1000 chance of losing the $2 bet. Assume that it costs $2 to play a lottery. The player chooses a three-digit number between 000 and 999, inclusive, and if the number is selected, then the player wins $500 ($498 considering the $2 cost to play). What is the expected value of this game (to the nearest cent)? If you are betting on a single 3-digit number (there are 1000 of them), then you have a probability of 1 / 1000 of winning $498. Amt won or lostProbability Winnings x Probability $498 1 / 1000 ─ $2 998 / 1000

112. MATH 110 Sec 13-4 Lecture: Expected Value And that means that you have a 999 / 1000 chance of losing the $2 bet. Assume that it costs $2 to play a lottery. The player chooses a three-digit number between 000 and 999, inclusive, and if the number is selected, then the player wins $500 ($498 considering the $2 cost to play). What is the expected value of this game (to the nearest cent)? If you are betting on a single 3-digit number (there are 1000 of them), then you have a probability of 1 / 1000 of winning $498. Amt won or lostProbability Winnings x Probability $498 1 / 1000 ─ $2 999 / 1000

113. MATH 110 Sec 13-4 Lecture: Expected Value And that means that you have a 999 / 1000 chance of losing the $2 bet. Amt won or lostProbability Winnings x Probability $498 1 / 1000 ─ $2 999 / 1000 Assume that it costs $2 to play a lottery. The player chooses a three-digit number between 000 and 999, inclusive, and if the number is selected, then the player wins $500 ($498 considering the $2 cost to play). What is the expected value of this game (to the nearest cent)? If you are betting on a single 3-digit number (there are 1000 of them), then you have a probability of 1 / 1000 of winning $498.

114. MATH 110 Sec 13-4 Lecture: Expected Value And that means that you have a 999 / 1000 chance of losing the $2 bet. Amt won or lostProbability Winnings x Probability $498 1 / 1000 ─ $2 999 / 1000 Assume that it costs $2 to play a lottery. The player chooses a three-digit number between 000 and 999, inclusive, and if the number is selected, then the player wins $500 ($498 considering the $2 cost to play). What is the expected value of this game (to the nearest cent)? If you are betting on a single 3-digit number (there are 1000 of them), then you have a probability of 1 / 1000 of winning $498.

115. MATH 110 Sec 13-4 Lecture: Expected Value And that means that you have a 999 / 1000 chance of losing the $2 bet. Assume that it costs $2 to play a lottery. The player chooses a three-digit number between 000 and 999, inclusive, and if the number is selected, then the player wins $500 ($498 considering the $2 cost to play). What is the expected value of this game (to the nearest cent)? If you are betting on a single 3-digit number (there are 1000 of them), then you have a probability of 1 / 1000 of winning $498. Amt won or lostProbability Winnings x Probability $498 1 / 1000 ─ $2 999 / 1000

116. MATH 110 Sec 13-4 Lecture: Expected Value And that means that you have a 999 / 1000 chance of losing the $2 bet. Assume that it costs $2 to play a lottery. The player chooses a three-digit number between 000 and 999, inclusive, and if the number is selected, then the player wins $500 ($498 considering the $2 cost to play). What is the expected value of this game (to the nearest cent)? If you are betting on a single 3-digit number (there are 1000 of them), then you have a probability of 1 / 1000 of winning $498. Amt won or lostProbability Winnings x Probability $498 1 / 1000 ─ $2 999 / 1000

117. MATH 110 Sec 13-4 Lecture: Expected Value And that means that you have a 999 / 1000 chance of losing the $2 bet. Assume that it costs $2 to play a lottery. The player chooses a three-digit number between 000 and 999, inclusive, and if the number is selected, then the player wins $500 ($498 considering the $2 cost to play). What is the expected value of this game (to the nearest cent)? If you are betting on a single 3-digit number (there are 1000 of them), then you have a probability of 1 / 1000 of winning $498. Amt won or lostProbability Winnings x Probability $498 1 / 1000 ─ $2 999 / 1000

118. MATH 110 Sec 13-4 Lecture: Expected Value And that means that you have a 999 / 1000 chance of losing the $2 bet. Assume that it costs $2 to play a lottery. The player chooses a three-digit number between 000 and 999, inclusive, and if the number is selected, then the player wins $500 ($498 considering the $2 cost to play). What is the expected value of this game (to the nearest cent)? If you are betting on a single 3-digit number (there are 1000 of them), then you have a probability of 1 / 1000 of winning $498. Amt won or lostProbability Winnings x Probability $498 1 / 1000 ─ $2 999 / 1000 Another way to look at it is that if you make this bet often, on average, you can expect to lose about $1.50 for every $2 bet.

119. MATH 110 Sec 13-4 Lecture: Expected Value If a game has an expected value of 0, we say that the game is FAIR because, on average, a player will neither win nor lose money.

120. MATH 110 Sec 13-4 Lecture: Expected Value If a game has an expected value of 0, we say that the game is FAIR because, on average, a player will neither win nor lose money. None of the games that we have looked at so far are fair.

121. MATH 110 Sec 13-4 Lecture: Expected Value If a game has an expected value of 0, we say that the game is FAIR because, on average, a player will neither win nor lose money. Let’s examine the previous game and see if we can find the ticket price that would make the game fair. None of the games that we have looked at so far are fair.

122. MATH 110 Sec 13-4 Lecture: Expected Value If a game has an expected value of 0, we say that the game is FAIR because, on average, a player will neither win nor lose money. None of the games that we have looked at so far are fair. Assume that it costs $2 to play a lottery. The player chooses a three-digit number between 000 and 999, inclusive, and if the number is selected, then the player wins $500 ($498 considering the $2 cost to play). What is the expected value of this game to the nearest cent? Let’s examine the previous game and see if we can find the ticket price that would make the game fair.

123. MATH 110 Sec 13-4 Lecture: Expected Value If a game has an expected value of 0, we say that the game is FAIR because, on average, a player will neither win nor lose money. None of the games that we have looked at so far are fair. Assume that it costs $2 to play a lottery. The player chooses a three-digit number between 000 and 999, inclusive, and if the number is selected, then the player wins $500 ($498 considering the $2 cost to play). What is the expected value of this game to the nearest cent? Let’s examine the previous game and see if we can find the ticket price that would make the game fair.

124. MATH 110 Sec 13-4 Lecture: Expected Value If a game has an expected value of 0, we say that the game is FAIR because, on average, a player will neither win nor lose money. None of the games that we have looked at so far are fair. Assume that it costs $2 to play a lottery. The player chooses a three-digit number between 000 and 999, inclusive, and if the number is selected, then the player wins $500 ($498 considering the $2 cost to play). What is the expected value of this game to the nearest cent? Actually, it is very easy to find the price to make this a fair game. Simply adjust the price of playing by the expected value (-$1.50). Let’s examine the previous game and see if we can find the ticket price that would make the game fair.

125. MATH 110 Sec 13-4 Lecture: Expected Value If a game has an expected value of 0, we say that the game is FAIR because, on average, a player will neither win nor lose money. None of the games that we have looked at so far are fair. Assume that it costs $2 to play a lottery. The player chooses a three-digit number between 000 and 999, inclusive, and if the number is selected, then the player wins $500 ($498 considering the $2 cost to play). What is the expected value of this game to the nearest cent? Actually, it is very easy to find the price to make this a fair game. Simply adjust the price of playing by the expected value (-$1.50). Let’s examine the previous game and see if we can find the ticket price that would make the game fair. So to make this game fair, the cost to play should be $2.00 - $1.50 = $0.50.

126. MATH 110 Sec 13-4 Lecture: Expected Value If a game has an expected value of 0, we say that the game is FAIR because, on average, a player will neither win nor lose money. None of the games that we have looked at so far are fair. Assume that it costs $2 to play a lottery. The player chooses a three-digit number between 000 and 999, inclusive, and if the number is selected, then the player wins $500 ($498 considering the $2 cost to play). What is the expected value of this game to the nearest cent? Actually, it is very easy to find the price to make this a fair game. Simply adjust the price of playing by the expected value (-$1.50). Let’s examine the previous game and see if we can find the ticket price that would make the game fair. So to make this game fair, the cost to play should be $2.00 - $1.50 = $0.50.