aez-notes

The following are two versions of the matching problem:

(a) From a shuffled deck, cards are laid out on a table one at a time, face up from left to right, and then another deck is laid out so that each of its cards is beneath a card of the first deck. What is the average number of matches of the card above and the card below in repetitions of experiment?

(b) A typist types letters and envelops to \(n\) different persons. The letters are randomly put into the envelopes. On the average, how many letters are put into their own envelopes?

Consider an indicator variable for a match at a particular comparison. The total number of matches is the sum of the indicators and expectation is linear, so we just need the probability of a match in a single instance. For the cards this is \(1/52\) and there are \(52\) cards so on the average there will be one match. It is the same for the letters.