Q: You meet a stranger on the street, and ask how many children he has. He truthfully says two. You ask "Is the older one a girl?" He truthfully says yes. What is the probability that both children are girls? What would the probability be if your second question had been "Is at least one of them a girl?", with the other conditions unchanged?

S: There are four equally-likely possibilities
Oldest child Youngest child

1. Girl Girl 2. Girl Boy 3. Boy Girl 4. Boy Boy

If the stranger says "My oldest child is a girl," he has eliminated cases 3 and 4, and in the remaining cases both are girls 1/2 of the time. If the stranger says "At least one of my children is a girl," he has eliminated case 4 only, and in the remaining cases both are girls 1/3 of the time.

In actuality, there are about 106 boys born for every 100 girls, and the odds of having a second child are influenced by the sex of the first, so these four possibilities are not really equally likely. According to a reader of the "Ask Marilyn" column (Parade magazine, October 19, 1997), the US Census Bureau surveyed 42,888 two-children families between 1987 and 1993. Of these, 11,334 (26.4%) were boy-boy, 11,118 (25.9%) were boy-girl, 10,913 (25.4%) were girl-boy and 9523 (22.2%) were girl-girl. Using these figures, if the oldest child is a girl, then the odds that they both are girls is 46.6%, whereas if at least one is a girl, the odds that they both are girls is 30.2%.

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