Exercises

Jun 09'24

Let [math]T[/math] be the random variable that counts the number of 2-unshuffles performed on an [math]n[/math]-card deck until all of the labels on the cards are distinct. This random variable was discussed in Card Shuffling. Using Equation in that section, together with the formula

[[math]] E(T) = \sum_{s = 0}^\infty P(T \gt s) [[/math]]

that was proved in Exercise, show that

[[math]] E(T) = \sum_{s = 0}^\infty \left(1 - {{2^s}\choose n}\frac {n!}{2^{sn}}\right)\ . [[/math]]

Show that for [math]n = 52[/math], this expression is approximately equal to 11.7. (As was stated in Combinatorics, this means that on the average, almost 12 riffle shuffles of a 52-card deck are required in order for the process to be considered random.)