Afamily 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?

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...

The solution is and

Step-by-step explanation:

The expression is

Multiplying the term by , we get,

Adding both sides by 1,

Taking factor, we get,

Equating the factors to 0,

and

Thus, the smaller value is

The solution is and