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Conditional Probability What is a conditional probability? • It is the probability of an event in a subset of the sample space • Example: Roll a die twice, win if total ≥ 9 • Sample space S = set of outcomes = {11, 12, 13, 14, 15, 16, 21, 22, …, 65, 66} • Event W = pairs that sum to ≥ 9 = {36, 45, 46, 54, 55, 56, 63, 64, 65, 66} • Pr(W) = 10/36 What is a conditional probability? • Now suppose we know that the first roll is 4 or 5. What is now the probability that the sum of the two rolls will be ≥ 9? • Let B = first roll is 4 or 5 = {41, 42, …, 46, 51, 52, …, 56} • Event W∩B = {45, 46, 54, 55, 56} • Pr(W | B) = |W∩B|/|B| = 5/12 • “Probability of W given B” Conditional probability • But since the sample space is the same, | W B | | W B | / | S | Pr(W B) Pr(W | B) |B| |B|/|S| Pr(B) • In general, the conditional probability of event A given event B is defined as Pr(A B) Pr(A | B) Pr(B) What is the difference between Pr(A|B) and Pr(B|A)? • Pr(A|B) is the proportion of B that is also within A, that is, Pr(A|B) is |A∩B| as a proportion of |B| A B A∩B • Pr(A|B) is close to 1 but Pr(B|A) is close to 0 CS20 • This class has 42 students, 13 freshmen, 17 women, and 5 women freshmen • So if a student is selected at random, – Pr(Freshman) = 13/42, – Pr(Woman) = 17/42 – Pr(Woman freshman) = 5/42. • If a random selection chooses a woman, what is the probability she is a freshman? – Simple way: #women freshmen/#women = 5/17 – Using probability: Pr(W F) 5 / 42 5 Pr(F | W ) Pr(W ) 17 / 42 17 Conditional Probability and Independence • Fact: A and B are independent events iff Pr(A|B) = Pr(A). • That is, knowing whether B is the case gives no information that would help determine the probability of A. • Proof: A and B independent iff Pr(A)∙Pr(B) = Pr(A∩B) Pr(A∩B) = Pr(A|B)∙Pr(B) So as long as Pr(B) is nonzero, Pr(A)∙Pr(B) = Pr(A|B)∙Pr(B) iff Pr(A) = Pr(A|B) Total Probability • Suppose (hypothetically!): – Rick Santorum has a 5% probability of getting enough delegates to become the Republican nominee, unless the voting goes beyond the first ballot and there is a brokered convention – In a brokered convention, Santorum has a 65% probability of winning the nomination – There is a 7% probability of a brokered convention (cf. Intrade.com) • What is the probability that Santorum will be the Republican nominee? Total Probability Simple version: For any events A and B whose probability is neither 0 nor 1: Pr(A) Pr(A | B)Pr(B) Pr(A | B)Pr(B) That is, Pr(A) is the weighted average of the probability of A conditional on B happening, and the probability of A conditional on B not happening. _ B B A S “Total probability” = weighted average of probabilities Pr(S) Pr(S | B)Pr(B) Pr(S | B)Pr(B) • Pr(Santorum|Brokered) = .65 • Pr(Santorum|¬Brokered) = .05 • Pr(Brokered) = .07 • Then Pr(Santorum) = .65∙.07 + .05∙.93 = .092 FINIS