Section 10.4

Positions in a Shuffled Deck

Expected position of the first ace or the last card of a suit via exchangeable gaps, (n+1)/(k+1).

Toolkit

Idea: in a uniformly random arrangement of $n$ cards, the $k$ “special” cards (aces, kings, etc.) divide the remaining $n-k$ cards into $k+1$ gaps that are exchangeable, hence equal in expectation. This gives a quick handle on expected positions and counts without summing indicator probabilities card by card.

Key results:

\mathbb{E}[\text{size of each gap}] = \frac{n-k}{k+1}, \qquad \mathbb{E}[\text{position of first special card}] = \frac{n+1}{k+1}.

More generally, with $s$ “ordinary” items randomly interspersed among $L$ “marker” items (so $L+1$ gaps), the expected number of ordinary items before the next marker is $s/(L+1)$ .

Useful techniques: exchangeability makes gap sizes independent of the relative order of the dividers, so one may condition on that order; linearity in the remaining deck size lets the formula be reapplied recursively (tower property) after conditioning on which type of special card appears first.

Problems