15. Three tired cowboys entered a saloon and hung their hats on a buffalo horn at the entrance. When the cowboys left deep into the night, they were unable to distinguish one hat from another, and therefore picked three hats at random. Find the probability that none of them took their own hat.
# Solution.

Note that the cowboys have 6 different ways to take the hats. Indeed, the first cowboy can take any of the three hats, the second one can take one of the remaining two, and the third will take the last hat.

At the same time, in only two out of the six scenarios, none of the cowboys will take their own hat, since the cowboy who takes the hat first can take any of the two hats belonging to the other cowboys. The remaining cowboys have no choice, as one of their hats is still not taken, and thus, he must take the other hat.

Thus, the desired probability is $\frac{2}{6}=\frac{1}{3}$.

Answer: $\frac{1}{3}$.