Random Walk: Boys and Girls

Consider the difference between the number of girls and boys as a simple symmetric random walk. Without loss of generality, assume the first child is a girl, establishing an initial difference of 1. The random walk proceeds by incrementing or decrementing this difference by 1 with equal probability at each step.

For a simple symmetric random walk on the integers starting at 1, the probability of eventually returning to 0 (i.e., having an equal number of boys and girls) is 1. However, the expected time to return to 0 is infinite. This implies that, on average, the couple will never achieve an equal number of boys and girls.