A couple will have children until they have two boys or a maximum of four children. The possible gender combinations are BB, GBB, GGBB, GGBG, GGG, GBGB, GBGG, BGGB, BGGG, BGB. The probability of a child being a boy or girl is 0.5. Part a asks to write a table with the probabilities and values of the random variable X, the number of boys. Part b asks to do the same for the random variable Y, the number of children.

1. 1. A couple decides to have children until they have two boys, but no more than four children.
The couple will also stop trying when it is clear that they will not have two boys (which is the
reason why GGG is an outcome and GGGB GGGG are not). The sample space S in this case is
therefore: S={BB,GBB, GGBB, GGBG, GGG, GBGB, GBGG, BGGB, BGGG, BGB } Assume
that a child is equally likely to be a boy or a girl [P(B)=P(G)=.5], and that the gender of one child
is independent of the gender of any other child. (a) Let the random variable X be the number of
boys that the couple will have. Write down the probability distribution function of X (in a form
of a table of possible values and probabilities). Guidance: As we did in Lecture #14 (page 2) and
Lecture #15 (page 1), it would be helpful to write above each outcome its probability, and below
each outcome the value of X it corresponds to. (b) Let the random variable Y be the number of
children that the couple will have. Write down the probability distribution function of Y (in a
form of a table of possible values and probabilities).