2. 1 Weather forecast Psychology Games Sports Chapter 3 Elementary Statistics Larson Farber Probability Business Medicine

3. 2 { 1 2 3 4 5 6 } { Die is even }={ 2 4 6 } {4} Roll a die Probability experiment: An action through which counts, measurements or responses are obtained Sample space: The set of all possible outcomes Event: A subset of the sample space. Outcome: The result of a single trial Important Terms

4. 3 Probability Experiment: An action through which counts, measurements, or responses are obtained Sample Space: The set of all possible outcomes Event: A subset of the sample space. Outcome: The result of a single trial Choose a car from production line Q Q Q   Q  Q Q   Q      Another Experiment 

5. 4 Classical (equally probable outcomes) Probability blood pressure will decrease after medication Probability the line will be busy Empirical Subjective Types of Probability

6. 5 Two dice are rolled. Describe the sample space. 1st roll 36 outcomes 2nd roll Tree Diagrams Start 123456 1 2 3 4 5 61 2 3 4 5 61 2 3 4 5 61 2 3 4 5 61 2 3 4 5 61 2 3 4 5 6

7. 6 1,1 1,2 1,3 1,4 1,5 1,6 2,12,1 2,22,2 2,32,3 2,42,4 2,52,5 2,6 3,1 3,2 3,3 3,4 3,5 3,6 4,1 4,2 4,3 4,4 4,5 4,6 5,1 5,2 5,3 5,4 5,5 5,6 6,1 6,2 6,3 6,4 6,5 6,6 Find the probability the sum is 4 Find the probability the sum is 11 Find the probability the sum is 4 or 11 Sample Spaces and Probabilities Two dice are rolled and the sum is noted.

8. 7 Complementary Events The complement of event E is event E. E consists of all the events in the sample space that are not in event E.  The day’s production consists of 12 cars, 5 of which are defective. If one car is selected at random, find the probability it is not defective. E E Solution: P(defective) = 5/12 P(not defective) = 1 - 5/12 = 7/12 = 0.583 P(E´) = 1 - P(E) Q

9. 8 The probability an event B will occur, given (on the condition) that another event A has occurred.  Two planes are selected from a production line of 12 planes where 5 are defective. What is the probability the 2nd plane is defective, given the first plane was defective? We write this as P(B|A) and say “probability of B, given A”. Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Given a defective plane has been selected, the conditional sample space has 4 defective out of 11. So, P(B|A) = 4/11 Q Conditional Probability

10. 9  Two dice are rolled, find the probability the second die is a 4, given the first was a 4. Original sample space : {1, 2, 3, 4, 5, 6 } Given the first die was a 4, the conditional sample space is : {1, 2, 3, 4, 5, 6} The conditional probability, P(B|A) = 1/6 Conditional Probability

11. 10 Two events A and B are independent if the probability of the occurrence of event B is not affected by the occurrence (or non-occurrence) of event A. A= taking an aspirin each day B= having a heart attack A= being a female B= being under 64” tall Two events that are not independent are dependent. A= Being female B=Having type O blood A= 1st child is a boy B= 2nd child is a boy Independent Events

12. 11 If events A and B are independent, then P(B|A) = P(B)  12 planes are on a production line where 5 are defective and 2 planes are selected at random. A= first plane is defective B= second plane is defective. The probability of getting a defective plane for the second plane depends on whether the first was defective. The events are dependent.  Two dice are rolled. A= first is a 4 and B = second is a 4 P(B)= 1/6 and P(B|A) = 1/6. The events are independent. Independent Events Conditional Probability Probability

13. 12 The results of responses when a sample of adults in 3 cities were asked if they liked a new juice is: 1. P(Yes) 2. P(Seattle) 3. P(Miami) 4. P(No, given Miami) 5. P(Not Seattle) 6. P(Seattle, given yes) 7. P(Yes, given Seattle) 8. P(Miami, given Omaha) OmahaSeattleMiamiTotal Yes100150 400 No12513095 350 Undecided 75170 5 250 Total3004502501000 Contingency Table One of the responses is selected at random. Find:

14. 13 1. P(Yes) 2. P(Seattle) 3. P(Miami) 4. P(No, given Miami) 100150 12513095 350 75170 5 250 OmahaSeattleMiamiTotal Yes No Und Total300450250 400 1000 = 400 / 1000 = 0.4 = 95 / 250 = 0.38 =250 / 1000 = 0.25 Solutions Answers: 1) 0.4 2) 0.45 3) 0.25 4) 0.38 = 450 / 1000 = 0.45

15. 14 100150 12513095 350 75170 5 250 OmahaSeattleMiamiTotal Yes No Undecided Total300450250 400 1000 = 1 - 450 / 1000 = 0.55 = 0 / 250 = 0 Solutions Answers: 5) 0.55 6) 0.375 7) 0.333 8) 0 5. P(Not Seattle) 6. P(Seattle, given yes) 7. P(Yes, given Seattle) 8. P(Miami, given Omaha) = 150 / 400 = 0.375 =150 / 450 = 0.333

16. 15 Are events A= Seattle and B= Yes independent events? Are events A = Miami and B = Omaha independent events? OmahaSeattleMiamiTotal Yes100150 400 No12513095 350 Undecided 75170 5 250 Total3004502501000 Solutions P(Yes |Seattle) = 150/450 = 0.333 P(Yes) = 0.4 If events are independent P(B|A) = P(B) Since 0.333  0.4 the events are NOT independent. P(Omaha|Miami) = 0 P(Omaha) = 0.3 Since 0  0.3 the events are NOT independent. If events are independent P(B|A) = P(B)

17. 16 To find the probability that two events, A and B will occur in sequence, multiply the probability A occurs by the conditional probability B occurs, given A has occurred. P( A and B) = P(A) × P(B|A)  Two planes are selected from a production line of 12 where 5 are defective. Find the probability both planes are defective. A = first plane is defective B = second plane is defective. P(A) = 5/12 P(B|A) = 4/11 P(A and B) = 5/12 × 4/11 = 5/33= 0.1515 Multiplication Rule Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q

18. 17  Two dice are rolled. Find the probability both are 4’s. A= first die is a 4 and B= second die is a 4. P(A) = 1/6P(B|A) = 1/6 P(A and B) = 1/6 × 1/6 = 1/36 = 0.028 When two events A and B are independent, then P (A and B) = P(A) × P(B) Note for independent events P(B) and P(B|A) are the same. Multiplication Rule

19. 18 Mutually Exclusive Events Two events, A and B are mutually exclusive, if they cannot occur in the same trial. A= A person is under 21 B= A person is running for the U.S. Senate A = A person was born in Philadelphia B = A person was born in Houston A B Mutually exclusive P(A and B) = 0 When event A occurs it excludes event B in the same trial.

20. 19 Non-Mutually Exclusive Events If two events can occur in the same trial, they are non- mutually exclusive. A = A person is under 25 B = A person is a lawyer A = A person was born in Philadelphia B = A person watches 20/20 on TV. A B Non-mutually exclusive P(A and B)  0 A and B

21. 20 The Addition Rule The probability that one or the other of two events will occur is: P(A) + P(B) - P(A and B)  A card is drawn from a deck. Find the probability it is a king or it is red. A= the card is a king B = the card is red. P(A) = 4/52 and P(B) = 26/52 but P( A and B) = 2/52 P(A or B) = 4/52 + 26/52 - 2/52 = 28/52 = 0.538   

22. 21 The Addition Rule  A card is drawn from a deck. Find the probability the card is a king or a 10. A = the card is a king and B = the card is a 10. P(A) = 4/52 and P(B) = 4/52 and P( A and B) = 0/52 P(A or B) = 4/52 + 4/52 - 0/52 = 8/52 = 0.054 When events are mutually exclusive, P(A or B) = P(A) +P(B)

23. 22 The results of responses when a sample of adults in 3 cities was asked if they liked a new juice is: Contingency Table 4. P(Miami or Yes) 5. P(Omaha or No) 6. P(Miami or Seattle) 1. P(Miami and Yes) 2. P(Omaha and No) 3. P(Miami and Seattle) OmahaSeattleMiamiTotal Yes100150 400 No12513095 350 Undecided 75170 5 250 Total3004502501000 One of the responses is selected at random. Find:

24. 23 Contingency Table 1. P(Miami and Yes) 2. P(Omaha and No) 3. P(Miami and Seattle) = 250/1000 * 150/250 = 150/1000 = 0.15 = 300/1000 * 125/300 = 125/1000 = 0.125 = 0 OmahaSeattleMiamiTotal Yes100150 400 No12513095 350 Undecided 75170 5 250 Total3004502501000 One of the responses is selected at random. Find:

25. 24 Contingency Table 4. P(Miami or Yes) 5. P(Omaha or No) 6. P(Miami or Seattle) 250/1000 + 450/1000 - 0/1000 =700/1000 = 0.7 Answers: 4) 0.5 5) 0.525 6) 0.7 OmahaSeattleMiamiTotal Yes100150 400 No12513095 350 Undecided 75170 5 250 Total3004502501000 250/1000 + 400/1000 - 150/1000 =500/1000 = 0.5 300/1000 + 350/1000 - 125/1000 =525/1000 = 0.525

26. 25 Summary Probability at least one of two events occur P(A or B) = P(A) + P(B) - P(A and B) Add the simple probabilities but to prevent double counting, don’t forget to subtract the probability of both occurring For complementary events P(E') = 1 - P(E) Subtract the probability of the event from one. The probability both of two events occur P(A and B) = P(A) *P(B|A) Multiply the probability of the first event by the conditional probability the second event occurs, given the first occurred.

27. 26 Fundamental Counting Principle If one event can occur m ways and a second event can occur n ways, the number of ways the two events can occur in sequence is m*n. This rule can be extended for any number of events occurring in a sequence.  If a meal consists of 2 choices of soup, 3 main dishes and 2 desserts, how many different meals can be selected? = 12 meals Start 2 Soup * 3 Main 2 * Dessert

28. 27 Permutations A permutation is an ordered arrangement The number of permutations for n objects is n! n! = n*(n - 1)*(n -2)…..3*2*1 The number of permutations of n objects taken r at a time is  You are required to read 5 books from a list of 8. In how many different orders can you do so? You have 6720 permutations of 8 books reading 5.

29. 28 Combinations A combination is an selection or r objects from a group of n objects. The number of combinations of n objects taken r at a time is  You are required to read 5 books from a list of 8. In how many different ways can you choose the books if order does not matter. There are 56 combinations of 8 objects taking 5.

30. 29 Permutations of 4 objects taking 2 Each of the 12 groups represents a permutation.

31. 30 Combinations of 4 objects taking 2 Each of the 6 groups represents a combination

32. 31 Distinguishable Permutations What if when we are selecting items, and order matters and some are the same? n= the total number of items n = the number of each choice Ex- How many ways can we order the letters of illinois?