2. Probability Formal study of uncertainty The engine that drives statistics Primary objective of lecture unit 4: use the rules of probability to calculate appropriate measures of uncertainty.

3. Introduction Nothing in life is certain We gauge the chances of successful outcomes in business, medicine, weather, and other everyday situations such as the lottery (recall the birthday problem)

4. History For most of human history, probability, the formal study of the laws of chance, has been used for only one thing: gambling

5. History (cont.) Nobody knows exactly when gambling began; goes back at least as far as ancient Egypt where 4-sided “astragali” (made from animal heelbones) were used

6. History (cont.) The Roman emperor Claudius (10BC-54AD) wrote the first known treatise on gambling. The book “How to Win at Gambling” was lost. Rule 1: Let Caesar win IV out of V times

7. Approaches to Probability Relative frequency event probability = x/n, where x=# of occurrences of event of interest, n=total # of observations Coin, die tossing; nuclear power plants? Limitations repeated observations not practical

8. Approaches to Probability (cont.) Subjective probability individual assigns prob. based on personal experience, anecdotal evidence, etc. Classical approach every possible outcome has equal probability (more later)

9. Basic Definitions Experiment: act or process that leads to a single outcome that cannot be predicted with certainty Examples: 1.Toss a coin 2.Draw 1 card from a standard deck of cards 3.Arrival time of flight from Atlanta to RDU

10. Basic Definitions (cont.) Sample space: all possible outcomes of an experiment. Denoted by S Event: any subset of the sample space S; typically denoted A, B, C, etc. Simple event: event with only 1 outcome Null event: the empty set  Certain event: S

11. Examples 1.Toss a coin once S = {H, T}; A = {H}, B = {T} simple events 2.Toss a die once; count dots on upper face S = {1, 2, 3, 4, 5, 6} A=even # of dots on upper face={2, 4, 6} B=3 or fewer dots on upper face={1, 2, 3}

12. Laws of Probability

13. Laws of Probability (cont.) 3.P(A’ ) = 1 - P(A) For an event A, A’ is the complement of A; A’ is everything in S that is not in A. A A' S

14. Birthday Problem What is the smallest number of people you need in a group so that the probability of 2 or more people having the same birthday is greater than 1/2? Answer: 23 No. of people23304060 Probability.507.706.891.994

15. Example: Birthday Problem A={at least 2 people in the group have a common birthday} A’ = {no one has common birthday}

16. Unions and Intersections S A B A  A 

17. Mutually Exclusive (Disjoint) Events Mutually exclusive or disjoint events-no outcomes from S in common S A B A  = 

18. Laws of Probability (cont.) Addition Rule for Disjoint Events: 4. If A and B are disjoint events, then P(A  B) = P(A) + P(B)

19. Laws of Probability (cont.) General Addition Rule 5. For any two events A and B P(A  B) = P(A) + P(B) – P(A  B)

20. P(A  B)=P(A) + P(B) - P(A  B) S AB A 

21. Example: toss a fair die once S = {1, 2, 3, 4, 5, 6} A = even # appears = {2, 4, 6} B = 3 or fewer = {1, 2, 3} P(A  B) = P(A) + P(B) - P(A  B) =P({2, 4, 6}) + P({1, 2, 3}) - P({2}) = 3/6 + 3/6 - 1/6 = 5/6

22. Laws of Probability: Summary 1. 0  P(A)  1 for any event A 2. P(  ) = 0, P(S) = 1 3. P(A’) = 1 – P(A) 4. If A and B are disjoint events, then P(A  B) = P(A) + P(B) 5. For any two events A and B, P(A  B) = P(A) + P(B) – P(A  B)

23. End of First Part of Section 4.1