## 14. Lacewing

One fine summer day, François was looking for Béatrice in Cabourg. Where could she be? Perhaps on the beach (one chance in two) or on the tennis court (one chance in four), or maybe in a café (also one chance in four). If Béatrice is on the beach, which is large and crowded, François has one chance in two of not finding her. If she is on one of the courts, there is another one chance in three of missing her, but if she went to the café, François will definitely find her: he knows exactly which café Béatrice usually indulges in ice cream. François searched all three possible meeting places, but he did not find Béatrice.

What is the probability that Béatrice was on the beach, assuming she did not change locations while François was looking for her?
14. The probability of meeting is:

on the beach $(1 / 2) \cdot(1 / 2)=1 / 4$,

on the court $(1 / 4) \cdot(2 / 3)=1 / 6$,

in the cafe $(1 / 4) \cdot(1)=1 / 4$.

The total probability of meeting is $4 / 6=2 / 3$.

Therefore, the probability that François will not find Béatrice is $1-2 / 3=1 / 3$.

However, the probability that they will miss each other on the beach is $(1 / 2) \cdot(1 / 2)=1 / 4$.

Thus, the probability that Béatrice was on the beach and François still did not find her is $(1 / 4):(1 / 3)=3 / 4$.