Section 3.1

Cards and Poker Hands

Probabilities of hands and deals from shuffled decks: poker hands, aces in a hand, pairs from a depleted deck, multi-deck draws.

Toolkit

These problems rest on the equally-likely-outcomes model: all $\binom{52}{k}$ (or $52!$ , or multinomial) deals are equally likely, so

\mathbb{P}(\text{event})=\frac{\#\text{favourable deals}}{\#\text{total deals}}.

Count favourable deals by a two-stage process: first choose the ranks involved (since suits/ranks among “equal-role” cards are interchangeable, choosing a set avoids overcounting), then choose suits or specific cards within those ranks. Useful tools:

Hypergeometric counting: choosing $a$ objects from one subgroup and $b$ from another, $\binom{m}{a}\binom{n}{b}$ , when sampling without replacement from a deck split into classes.
Multinomial deals: dealing a 52-card deck into hands of fixed sizes has $\binom{52}{n_1,n_2,\dots}$ equally likely outcomes.
Inclusion–exclusion for overlapping categories (e.g. straights that are also flushes): subtract the overcounted intersection.
When a few cards are removed from the deck, recompute counts on the remaining depleted composition before applying the above.

Problems