Solutions to homework 4

1) Lets list all possibilities;
BBB, BBG, BGB, GBB, GGB, GBG, BGG, GGG.
(i) P(A) = 7/8 (all except BBB)
(ii) P(B) = 1/8 (only GGG)
(iii) "A and B" is "they have at least one girl and they have exactly 3 girls". Clearly this is "they have exactly 3 girls", ie this is the same event as B.
(iv) P(A and B) = P(B) = 1/8.
(v) No, P(A and B) is not P(A)xP(B).
(vi) Knowing A has occured reduces the sample space to the 7 equally likely outcomes consisting of all outcomes except for BBB. Only one of these is B (GGG), so the probability of B given A is 1/7.

2)

Let's assume a large population of 10000, and arrange in a table below;

Note that according to this table, we have 392 people with positive results, and only 95 are infected, so the chance of being infected given a positive result is 95/392 = .242  or 24.2%.