Analysis of The Ambiguity
If it is assumed that this information was obtained by looking at both children to see if there is at least one boy, the condition is both necessary and sufficient. Three of the four equally probable events for a two-child family in the sample space above meet the condition:
| Older child | Younger child |
|---|---|
| Girl | Boy |
| Boy | Girl |
| Boy | Boy |
Thus, if it is assumed that both children were considered while looking for a boy, the answer to question 2 is 1/3. However, if the family was first selected and then a random, true statement was made about the gender of one child (whether or not both were considered), the correct way to calculate the conditional probability is not to count the cases that match. Instead, one must add the probabilities that the condition will be satisfied in each case:
| Older child | Younger child | P(this case) | P("at least one boy" given this case) | P(both this case, and "at least one boy") |
|---|---|---|---|---|
| Girl | Girl | 1/4 | 0 | 0 |
| Girl | Boy | 1/4 | 1/2 | 1/8 |
| Boy | Girl | 1/4 | 1/2 | 1/8 |
| Boy | Boy | 1/4 | 1 | 1/4 |
The answer is found by adding the numbers in the last column wherever you would have counted that case: (1/4)/(0+1/8+1/8+1/4)=1/2. Note that this is not necessarily the same as reporting the gender of a specific child, although doing so will produce the same result by a different calculation. For instance, if the younger child is picked, the calculation is (1/4)/(0+1/4+0+1/4)=1/2. In general, 1/2 is a better answer any time a Mr. Smith with a boy and a girl could have been identified as having at least one girl.
Read more about this topic: Boy Or Girl Paradox
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