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Given two events, A and B, Pr(A and B) = Pr(A)*Pr(B) if A and B are independent and Pr(A and B) = Pr(A | B)*Pr(B) if they are not.
Pr(3H given >= 2H) = Pr(3H and >= 2H)/Pr(>=2H) = Pr(3H)/Pr(>=2H) = (1/4)/(11/16) = 4/11.
Pr(Two different numbers) = 1 - Pr(Two same) = 1 - 1/6 = 5/6 = 83.3%
Suppose there is an event A and the probability of A happening is Pr(A). Then the complementary event is that A does not happen or that "not-A" happens: this is often denoted by A'.Then Pr(A') = 1 - Pr(A).Suppose there is an event A and the probability of A happening is Pr(A). Then the complementary event is that A does not happen or that "not-A" happens: this is often denoted by A'.Then Pr(A') = 1 - Pr(A).Suppose there is an event A and the probability of A happening is Pr(A). Then the complementary event is that A does not happen or that "not-A" happens: this is often denoted by A'.Then Pr(A') = 1 - Pr(A).Suppose there is an event A and the probability of A happening is Pr(A). Then the complementary event is that A does not happen or that "not-A" happens: this is often denoted by A'.Then Pr(A') = 1 - Pr(A).
The fact that the two people are picking alternately first from one class and then from the second means that the first picker always has first selection from whoever is remaining in each class; thus for the first picker: pr(1st girl) = 12/(12+13) = 12/25 Pr(2nd girl) = 11/(11+10) = 11/21 → Pr(two girls picked) = pr(1st girl) × pr(2nd girl) = 12/25×11/21 = 44/175