## 42. Twins

Anna and Brigitte, two little girls, are twins; they look like each other as much as two drops of water. At school, they sit at the same desk; one of them always sits on the left, and the other on the right. Each of them claims that she is Brigitte. The teacher knows from experience that, generally speaking, the girl on the right lies one time out of four, while the girl on the left lies one time out of five. But the teacher is interested in the probability that Anna is sitting on the left. Help her.
42. If Anna is sitting on the left and both girls call themselves Brigitte, this means that the girl sitting on the left[^24] is lying, while the girl on the right is telling the truth; the probability of this is

$$
1 / 5 \cdot 3 / 4=3 / 20
$$

But since both of them call themselves Brigitte, one of them is definitely lying, and the other is telling the truth; the probability of this is

$$
1 / 5 \cdot 3 / 4 + 4 / 5 \cdot 1 / 4 = 7 / 20
$$

The desired probability is the conditional probability of the event "Anna is sitting on the left" given that both girls call themselves Brigitte; it is equal to

$$
3 / 20 : 7 / 20 = 3 / 7
$$