[Math] Out of 10 people whats the probability that none gets his own umbrella…


In a party there are 10 people with same umbrellas,at the end of the party they get one umbrella randomly.What is the probability that none gets its own umbrella?
I was thinking on solving this in the followin way:
The probability of the first person picking a wrong umbrella is $\frac{9}{10} $, then for the second guy it will be $\frac{8}{9}$, for the third $\frac {7}{8}$ and so on,so the probability of event A to happen would be:
$$p(A) = \frac{9}{10} \cdot \frac{8}{9} \cdot \frac{7}{8} \cdots \frac{1}{2} \cdot 1$$
Is this solution correct,does the possibility of the first person getting the second person hats affect this solution?

Best Answer

The solution you give is not correct, and your comment at the end is the key to why. The first person does indeed have a $\frac9{10}$ chance of getting the wrong umbrella. However, the second person has a $\frac11$ probablity of getting the wrong umbrella if the first person picked his umbrella. Otherwise, if the first person picked someone else's umbrella, there would be an $\frac89$ chance that the second person would pick the wrong umbrella.

The proper way to look at this kind of question is to look at Derangements. The correct probability is $\frac{1334961}{3628800}=\frac{16481}{44800}$.