EXAMPLE 4 We want a girl

A couple plan to have children until they have a girl or until they have three children, whichever comes first. What is the probability that they will have a girl among their children?

Step 1. The probability model is like that for coin tossing:

  • • Each child has probability 0.49 of being a girl and 0.51 of being a boy. (Yes, more boys than girls are born. Boys have higher infant mortality, so the sexes even out soon.)

  • • The sexes of successive children are independent.

Step 2. Assigning digits is also easy. Two digits simulate the sex of one child. We assign 49 of the 100 pairs to “girl” and the remaining 51 to “boy”:

00, 01, 02, . . . , 48 = girl

49, 50, 51, . . . , 99 = boy

453

Step 3. To simulate one repetition of this childbearing strategy, read pairs of digits from Table A until the couple have either a girl or three children. The number of pairs needed to simulate one repetition depends on how quickly the couple get a girl. Here are 10 repetitions, simulated using line 130 of Table A. To interpret the pairs of digits, we have written G for girl and B for boy under them, have added space to separate repetitions, and under each repetition have written “+” if a girl was born and “−” if not.

6905 16 48 17 8717 40 9517 845340 648987 20
B G G G G B G G B G B B G B B B G
+ + + + + + + + − +

In these 10 repetitions, a girl was born nine times. Our estimate of the probability that this strategy will produce a girl is, therefore,

estimated probability

Some mathematics shows that, if our probability model is correct, the true probability of having a girl is 0.867. Our simulated answer came quite close. Unless the couple are unlucky, they will succeed in having a girl.