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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. 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) History • For most of human history, probability, the formal study of the laws of chance, has been used for only one thing: gambling 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 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 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 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) 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 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 F Certain event: S 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} Laws of Probability 1. 0 P ( A) 1, for any event A 2. P (F ) 0, P ( S ) 1 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. S A' A 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 people 23 30 40 60 Probability .507 .706 .891 .994 Example: Birthday Problem • A={at least 2 people in the group have a common birthday} • A’ = {no one has common birthday} 3 people 23 people :P ( A') 364 363 365 365 : 364 363 343 P ( A') . 498 365 365 365 so P ( A ) 1 P ( A ' ) 1 . 498 . 502 Unions and Intersections S A B A A Mutually Exclusive (Disjoint) Events • Mutually exclusive or disjoint events-no outcomes from S in common A = S A B 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) 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) P(AB)=P(A) + P(B) - P(A B) S A A B 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 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) End of First Part of Section 4.1