Question

A family decides to have children until it has three children of the same gender. Assuming P(B)=P(G)=.5,. what is the pmf of X = the number of children in the family?. This problem was in the section of negative binomial distribution in the textbook and i used the equation for nb but the sum of all the probabilities is just 0.5. And, i saw in yahoo someone using the P(a)= N(a)/N and getting the sum of P(x)=1. Also, it made sense for this problem as both the events were equally likely-- P(B)=p(G)=0.5. . But, shouldn't nb also give the same result as of using the classical definition?

Answer 1

The reason why the sum of your probabilities equals 0.5 is because you're only considering half of the problem--i.e. the probabilities for only one gender. For example, the probability that there are only three children and all are boys is 1/8. But, remember, it could be the case that there are three children and all are girls. Thus, the probability that there are three children is 1/4--not 1/8. Thus, multiply your result for P(X=4) and P(X=5) by 2 and you should end up with the right answer...