A couple decide they really want a daughter So they decide t

A couple decide they really want a daughter. So, they decide to start having children and continue until they have their first daughter. Assuming having either a boy or girl is equally likely, answer the following: In the end, will the couple he more likely to have more boys or more girls? Explain why. Give a formula for the probability that they end up with exactly k boys. Now assume that the probability of having a boy is 60%. Reanswer parts (a) and (b).

Solution

first note that the number of children is a geometric random variable (do it until you get the first success). if the probability of success is p, then the expected number of trials is 1/p. these facts are sufficient to answering these questions. (a) p=0.5. there is only one case where they will have girl more than boys, which is when the first child is a girl. the probability of that is p = 0.5. more boys and more girls are equally likely. (b) use geometric distribution p^k * (1-p) boy happens k times with probability p independently. and they get one girl at the end with probability (1-p) (c) now assume p=0.6. same logic. the probability of the first child being a girl = 0.6. the probability of the first child NOT being a girl = 1-0.6 = 0.4. having more girls is more likely. for (b), the answer is the same substitute p for 0.6.