## 9. A Group of Young People

The group consists of three boys and three girls. Each of the boys is loved by one of the three girls, and each girl is loved by one of the boys; but one of the girls sadly noted that in their group, no one is loved by the person they love themselves. Is this sad circumstance really so unlikely if we assume that both boys and girls choose their affection more or less randomly, at random?
9. Let's denote the girls by the letters $F_{1}, F_{2}, F_{3}$, and the three guys by $-G_{1}, G_{2}, G_{3}$. We will calculate the desired probability using the scheme («tree») illustrated in the figure.

The total number of outcomes corresponding to the "sad variant" of unrequited love turns out to be $3 \cdot 52=156$. The total number of possibilities is $3^{6}=729$. Therefore, the probability of the "sad outcome" is $156 / 729=0.214$. Alas! - it is not so small (more than 1 in 5).