1. Section 6.1
Day 3

2. Find and interpret the expected value for this lottery game
Find and interpret the expected value for this lottery game. A ticket costs $1.
Winnings, x Probability, p
$1 1/10
$2 1/14
$3 1/24
$ /200
$ /389
$ /20,000
$ /120,000

3. E(X) = $2.72?
Find and interpret the expected value for this lottery game. A ticket costs $1.
Winnings, x Probability, p
$1 1/10
$2 1/14
$3 1/24
$ /200
$ /389
$ /20,000
$ /120,000

4. Think again!!
Find and interpret the expected value for this lottery game. A ticket costs $1.
Winnings, x Probability, p
$1 1/10
$2 1/14
$3 1/24
$ /200
$ /389
$ /20,000
$ /120,000

5. Think again!!
Find and interpret the expected value for this lottery game. A ticket costs $1.
Winnings, x Probability, p
$1 1/10
$2 1/14
$3 1/24
$18 1/200
$50 1/389
$ /20,000
$ /120,000
Total

6. Expected Value
Find and interpret the expected value for this lottery game. A ticket costs $1.
Winnings, x Probability, p $0
$ /10
$ /14
$ /24
$ /200
$ /389
$ /20,000
$ /120,000

7. Expected Value
Find and interpret the expected value for this lottery game. A ticket costs $1.
Winnings, x Probability, p $
$ /10
$ /14
$ /24
$ /200
$ /389
$ /20,000
$ /120,000

8. E(X) = $0.6014
How do we interpret this expected value?

9. E(X) = $0.6014
How do we interpret this expected value?
If we spend $1, we expect to get back $
Or, the state can expect to pay out $ for every $1000 of tickets sold.

10. Linear Transformation Rule
Suppose you have a probability distribution with random variable X, mean x, and standard deviation x.

11. Linear Transformation Rule
Suppose you have a probability distribution with random variable X, mean x, and standard deviation x.
If you transform each value of x by multiplying it by d and then adding c, where c and d are constants, then

12. Linear Transformation Rule
Suppose you have a probability distribution with random variable X, mean x, and standard deviation x. If you transform each value by multiplying it by d and then adding c, where c and d are constants, then
c + dx = c + d x
c + dx = |d|● x

13. Expected Value
Find and interpret the expected value for this new lottery game. A ticket costs $1.
Winnings, x New winnings, 3x Probability, p
$0 $
$1 $3 1/10
$2 $6 1/14
$3 $9 1/24
$18 $54 1/200
$50 $ /389
$150 $ /20,000
$900 $ /120,000

14. New x = 0 + 3x, so c = 0 and d = 3
Find and interpret the expected value for this new lottery game. A ticket costs $1.
Winnings, x New winnings, 3x Probability, p
$0 $
$1 $3 1/10
$2 $6 1/14
$3 $9 1/24
$18 $54 1/200
$50 $ /389
$150 $ /20,000
$900 $ /120,000

15. New x = 0 + 3x, so c = 0 and d = 3 μx = 0.6014 for original game
We expect to win $1.804 for each dollar we spend.

16. Addition and Subtraction Rules
If X and Y are random variables, then
X + Y = X + Y
X - Y = X Y

17. Addition and Subtraction Rules
If X and Y are random variables, then
X + Y = X + Y
X - Y = X Y
and, if X and Y are independent, then
2X + Y = X Y
2X - Y = X Y

18. For each million tickets sold, the original New York lottery awarded one $50,000 prize, nine $5000 prizes, ninety $500 prizes, and nine hundred $50 prizes.
a. What was the expected value of a ticket?

19. x p
,000/1,000,000
/1,000,000
/1,000,000
/1,000,000
50,000 1/1,000,000

20. Expected value of a ticket is $0.185

21. Expected value of a ticket is $0.185
The tickets sold for $0.50 each. How much could the state of New York expect to earn for every million tickets sold?

22. Expected value of a ticket is $0.185
The tickets sold for $0.50 each. How much could the state of New York expect to earn for every million tickets sold?
1,000,000(0.50 – 0.185) = $315,000

23. Page 377, P7

24. Page 377, P7 Claire Charlotte Max Alisa Shaun
List all possible random samples of size 3 from this group of 5 students.

25. Page 377, P7 Claire Charlotte Max Alisa Shaun
List all possible random samples of size 3 from this group of 5 students.
5C3 = 10

26. Page 377, P7 Claire, Charlotte, Max Claire, Charlotte, Alisa
Claire, Charlotte, Shaun
Claire, Max, Alisa;
Charlotte, Alisa, Shaun
Claire, Max, Shaun;
Max, Alisa, Shaun
Claire, Alisa, Shaun
Charlotte, Max, Alisa
Charlotte, Max, Shaun

27. Page 377, P7

28. Page 377, P7 Claire, Charlotte, Max Claire, Charlotte, Alisa
Claire, Charlotte, Shaun
Claire, Max, Alisa;
Charlotte, Alisa, Shaun
Claire, Max, Shaun;
Max, Alisa, Shaun
Claire, Alisa, Shaun
Charlotte, Max, Alisa
Charlotte, Max, Shaun

29. Page 377, P7

30. Page 377, P10

31. Page 377, P10
How many possible samples of size 2?

32. Page 377, P10
How many possible samples of size 2?
6C2 = 15

33. Page 377, P10 (a) The 15 possible samples of size 2 are:
1 and 2; 1 and 3; 1 and 4; 1 and 5; 1 and 6
2 and 3; 2 and 4; 2 and 5; 2 and 6
3 and 4; 3 and 5; 3 and 6
4 and 5; 4 and 6
5 and 6

34. Page 377, P10
Assume computers 1, 2, and 3 are the defective monitors. (The probabilities would be the same no matter which 3 were assigned as the defective monitors).

35. Page 377, P10
Assume computers 1, 2, and 3 are the defective monitors. (The probabilities would be the same no matter which 3 were assigned as the defective monitors).
1 and 2; 1 and 3; 1 and 4; 1 and 5; 1 and 6
2 and 3; 2 and 4; 2 and 5; 2 and 6
3 and 4; 3 and 5; 3 and 6
4 and 5; 4 and 6
5 and 6

36. Page 377, P10

37. Page 377, P10

38. Page 377, P10

39. Page 381, E16

40. Page 381, E16

41. Page 381, E16

42. Page 381, E16

43. Page 381, E16

44. Page 381, E16

45. Page 382, E20

46. Page 382, E20

47. Page 382, E20

48. Questions?

49. Fathom Activity 6.1a

50. Activity 6.1a
x x
Die 1
Die 2
Sum
Difference

51. Activity 6.1a
x x
Die
Die 2
Sum
Difference

52. Activity 6.1a
x x
Die
Die
Sum
Difference

53. Fathom Activity 6.1a

54. Fathom Activity 6.1a

55. Activity 6.1a x 2x Die 1 3.5 2.917 Die 2 3.5 2.917 Sum 7 5.834
Difference

56. Activity 6.1a x 2x Die 1 3.5 2.917 Die 2 3.5 2.917 Sum 7 5.834
Difference

57. Activity 6.1a, 500 rolls
Variance
5.276
5.895

58. Fathom Activity 6.1a

59. Fathom Activity 6.1a