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Conditional Probability

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

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Conditional Probability

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  1. Conditional Probability

  2. 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

  3. 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”

  4. Conditional probability But since the sample space is the same, In general, the conditional probability of event A given event B is defined as

  5. What is the difference betweenPr(A|B) and Pr(B|A)? A B A∩B 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

  6. 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:

  7. Conditional Probability and Independence • Fact: A and B are independent events iffPr(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 iffPr(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) iffPr(A) = Pr(A|B)

  8. 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?

  9. Total Probability _ B B A S 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.

  10. “Total probability” = weighted average of probabilities Pr(Santorum|Brokered) = .65 Pr(Santorum|¬Brokered) = .05 Pr(Brokered) = .07 Then Pr(Santorum) = .65∙.07+ .05∙.93= .092

  11. FINIS

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