Demontmort’s Matching Problem with intuitive explanation and simulation.

Imagine the following scenario:

In a country wedding in France, every person removes their hat and places it on the hook before entering the lobby. After the wedding, each guest randomly grabs up a hat and leaves without looking over. What are the chances that atleast one guest has their own hat? This problem is similar to the Card Matching problem:

$Q.1$ In a well shuffled deck of cards labelled $1$ to $n$, what is the probability that atleast one card matches? A match is considered when the $i^{th}$ card in the deck is numbered $i$.

$Q.2$ Will the probability increase or decrease if the number of cards are increased?

The problem is the classic case of Inclusion-Exclusion principle. Before understanding about the principle, let’s try to simulate this problem on a deck of 20 cards and see the experimental result. We see as the number of simulation increases, the probability of atleast one match converges to approx. 0.63. Hence, there is around 63% chance that atleast one card will match in the deck or in the case of original scenario: atleast one guest will have their own hat after the wedding. Now let’s do some calculations to see if we can arrive at the same result.

# Inclusion-Exclusion Principle

Consider three events A, B, C. Can we write the probability of the union of these three sets as:

$P( A \cup B \cup C) = P(A) + P(B) + P(C)$

Did we add too much? Yes. [Fig.1] The intersection of the events have been double counted so we need to subtract (exclude) those from the equation.

$P( A \cup B \cup C)= P(A) + P(B) + P(C) - P(A \cap B) - P(B \cap C) - P(C \cap A)$

Did we subtract too much? Yes. [Fig. 2]

The intersection of the three events have been completely subtracted so we need to add (include) that to the equation. See the pattern? (include-exclude-include-….)

$P( A \cup B \cup C)= P(A) + P(B) + P(C) - P(A \cap B) - P(B \cap C) - P(C \cap A) + P(A \cap B \cap C)$

This is referred as the inclusion-exclusion principle. The generalized form of the equation can now be written as:

$P(A_1 \cup A_2 \ldots \cup A_n) = \sum_{j=1}^n P(A_j) - \sum_{i<j} P(A_i \cap A_j) + \sum_{i<j<k} P(A_i \cap A_j \cap A_k) - \ldots (-1)^{n+1} P(A_1 \cap A_2 \ldots \cap A_n)$

Now, lets come back to our initial card matching problem.

$P(\text{Atleast one card matches})= P(A_1 \cup A_2 \ldots \cup A_n)= P(\text{One card matches}) - P(\text{Two card matches})+ \ldots - P(\text{All card match})$

Calculating each term individually.

• $P(A_j)$: Probability that the $j^{th}$ card in the deck is $j$. As all cards are equally likely to occur:
$P (A_j) = \frac{1}{n}$

$\implies P(\text{One card matches}) = n* P(A_j) = n*\frac{1}{n}$ as there are n cards.

• $P(A_1 \cap A_2)$: Probability that the $1^{st}$ and $2^{nd}$ card in the deck match i.e. they are labeled $1$ and $2$ respectively.
$P (A_1 \cap A_2) = \frac{(n-2)!}{n!} = \frac{1}{n(n-1)}$

$\implies P(\text{Two card matches}) = (\text{No. of ways to choose two cards})* P(A_1 \cap A_2) = {n \choose 2} *\frac{1}{n(n-1)}$

• Similarly,
$P(A_1 \cap A_2 \cap \ldots A_k) = \frac{1}{n(n-1)\ldots(n-k+1)}$ $\implies P( k \text{ card matches}) = (\text{No. of ways to choose } k \text{ cards})* P(A_1 \cap A_2 \cap \ldots A_k) = {n \choose k} *\frac{1}{n(n-1)\ldots(n-k+1)}$

Finally,

\begin{aligned} P(\text{Atleast one card matches}) & = n*\frac{1}{n} - {n \choose 2} *\frac{1}{n(n-1)} + \ldots (-1)^{n+1}{n \choose n}*\frac{1}{n!} \\ & = n*\frac{1}{n} - \frac{n(n-1)}{2!} *\frac{1}{n(n-1)} + \ldots (-1)^{n+1}\frac{1}{n!} \\ & = 1 - \frac{1}{2!} + \frac{1}{3!}\ldots (-1)^{n+1}\frac{1}{n!} \\ & = 1-\frac{1}{e} = 0.63 \end{aligned}

Hence, there is a 63% chance that atleast one card matches. As we can see from the result, this chance does not depend on the number of cards in the deck contrary to our intuition that the chances of matching would change as the number increases.

 Blitzstein, Joseph K., and Jessica Hwang. Introduction to probability. Crc Press, 2019.