Discrrete mathematics for computer science conditional

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 ≥ Sample space S = set of outcomes = {11, 12, 13, 14, 15, 16, 21, 22, …, 65, 66} Event W = pairs that sum to ≥ = {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 or What is now the probability that the sum of the two rolls will be ≥ 9? Let B = first roll is or = {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, | Ω ∩ Β | | Ω ∩ Β | / | Σ | Πρ(Ω ∩ Β) Pr(W | B) = = = | Β| | Β| / | Σ| Πρ( Β) • In general, the conditional probability of event A given event B is defined as Πρ( Α ∩ Β) Pr(A | B) = Πρ( Β) 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 Pr(A|B) is close to but Pr(B|A) is close to A∩B CS20 • • • This class has 42 students, 13 freshmen, 17 women, and 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: Πρ(Ω ∩ Φ) / 42 Pr(F | W ) = = = Πρ(Ω ) 17 / 42 17 Conditional Probability and Independence • • Fact: A and B are independent events iff Pr(A|B) = Pr(A) • Proof: That is, knowing whether B is the case gives no information that would help determine the probability of A 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 nor 1: Pr(A) = Πρ( Α | Β)⋅ Πρ( Β) + Πρ( Α | Β)⋅ Πρ( Β) 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) = Πρ(Σ | Β)⋅ Πρ( Β) + Πρ(Σ | Β)⋅ Πρ( Β) • • • • Pr(Santorum|Brokered) = 65 Pr(Santorum|¬Brokered) = 05 Pr(Brokered) = 07 Then Pr(Santorum) = 65∙.07 + 05∙.93 = 092 FINIS ... of W given B” Conditional probability • But since the sample space is the same, | Ω ∩ Β | | Ω ∩ Β | / | Σ | Πρ(Ω ∩ Β) Pr(W | B) = = = | Β| | Β| / | Σ| Πρ( Β) • In general, the conditional probability... Πρ(Ω ) 17 / 42 17 Conditional Probability and Independence • • Fact: A and B are independent events iff Pr(A|B) = Pr(A) • Proof: That is, knowing whether B is the case gives no information that... Simple version: For any events A and B whose probability is neither nor 1: Pr(A) = Πρ( Α | Β)⋅ Πρ( Β) + Πρ( Α | Β)⋅ Πρ( Β) That is, Pr(A) is the weighted average of the probability of A conditional