# Expected number of cards in original position in a shuffled deck of $52$ cards?

Assume is shuffle is quite good that it randomizes the card order.

We know that E = $$\sum_{X=1}^n X*P(X)$$

We are already know that n=52 and that there are 52! ways to arrange the cards.

So probability that exactly 1 card is in correct position is $$\frac{1}{52!} {52 \choose 1}*$$(derangements of remaining cards)

This will be summed over all the 52 cases. This seems a bit complicated. Is there a simpler way?

• It would be interesting to find the probability that $n$ cards return to the same position. This seems harder to me. It would probably be approximately Poisson. – Ben Crowell Jul 5 at 18:39
• @BenCrowell Interesting question. Note special cases such as P(n=51)=0, since you can't have a single card out of place. – Ken Shirriff Jul 5 at 23:04
• @BenCrowell that would be a great separate question – qwr Jul 5 at 23:07

Let $$X_i$$ be a binary random variable that is $$1$$ if card $$i$$ being shuffled back to its original position, otherwise $$0$$.

We see $$E[X_i] = 1/52$$ since a card has an equal chance of each of 52 positions where it could be permuted to.

The magic step: the quantity we are looking for is $$E[X_1 + \cdots + X_{52}] = E[X_1] + \cdots + E[X_{52}] = 52 \times 1/52 = 1$$ by linearity of expectation!

(This is effectively the same answer as by Gribouillis, just in the language of expectation instead of computing all cases explicitly)

Examine the $$6$$ permutations of $$123$$:

$$\color{red}1\color{red}2\color{red}3$$ $$\color{red}1\color{black}3\color{black}2$$ $$\color{black}2\color{black}1\color{red}3$$ $$\color{black}2\color{black}3\color{black}1$$ $$\color{black}3\color{black}1\color{black}2$$ $$\color{black}3\color{red}2\color{black}1$$

with red indicating a correct position.

Then each number is in its correct position exactly twice, and there are three numbers, hence six red numbers, which gives the expected number of correct positions as $$\frac66=1$$.

In general there are $$n$$ numbers, and each is correct $$(n-1)!$$ times, hence $$n!$$ correct positions, and so the expected number of correct positions is $$\frac{n!}{n!}=1$$.

For $$i\in {1,..,n}$$ and $$\sigma$$ a random permutation, let $$\chi_{i=\sigma(i)}$$ have value $$1$$ if $$i=\sigma(i)$$ and zero otherwise. The expected number of cards returning in their initial position is $$$$m = \frac{1}{n!}\sum_{\sigma}\sum_{i=1}^n \chi_{i = \sigma(i)} = \frac{1}{n!}\sum_{i=1}^n \sum_\sigma \chi_{i=\sigma(i)} = \frac{1}{n!}\sum_{i=1}^n (n-1)! = 1$$$$ because $$(n-1)!$$ permutations leave the $$i$$-th element invariant.