Consider a pack of cards labelled $1, \ldots, 52$. We repeatedly take the top card and insert it uniformly at random in one of the 52 possible places, that is, either on the top or on the bottom or in one of the 50 places inside the pack. How long on average will it take for the bottom card to reach the top?

Let $p_{n}$ denote the probability that after $n$ iterations the cards are found in increasing order. Show that, irrespective of the initial ordering, $p_{n}$ converges as $n \rightarrow \infty$, and determine the limit $p$. You should give precise statements of any general results to which you appeal.

Show that, at least until the bottom card reaches the top, the ordering of cards inserted beneath it is uniformly random. Hence or otherwise show that, for all $n$,

$$\left|p_{n}-p\right| \leqslant 52(1+\log 52) / n$$