Hat Party!

Problem created by Tao Steven Zheng (郑涛)

Problem

Suppose you and five of your friends each brought a hat to a party. The hats are then put into a large box for a random hat draw. What is the probability that nobody selects his or her own hat?

Figure 1. Sesame street party hats

Solution

There are six people in total (you + 5 friends), so the number of possibilities that each person can get their hat is:

$$P_{6}=6!=720.$$

The number of possibilities that nobody selects their own hat is called the derangement problem. For six people, the number of derangements is:

$$D_{6}=6!\sum _{k=0}^{6}{\frac {{\left(-1\right)}^{k}}{k!}}=265$$

so the probability of each person get their wrong hat is:

$$p={\frac {265}{720}}$$

$$p={\frac {53}{144}}\approx 0.368056.$$

Therefore, the probability that nobody selects his or her own hat is 36.81%.

Note that a faster way of calculating the number of derangements is

$$D_{n}=\left[{\frac {n!}{e}}\right].$$