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, In general, the conditional probability of event A given event B is defined as

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| Pr(A|B) is close to 1 but Pr(B|A) is close to 0 AB A∩B

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:

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: 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(Santorum|Brokered) =.65 Pr(Santorum|¬Brokered) =.05 Pr(Brokered) =.07 Then Pr(Santorum) =.65∙ ∙.93 =.092

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