Counting under constraints

Apply required and excluded categories without counting the same hand twice.

12 minutes · intermediate · Original Poker4Quant lesson

What you will learn

Translate exact category constraints into products of combinations and conditional denominators.

Prerequisites: Pocket-hand counting · Hypergeometric models

Exactly one excludes the remaining members

For exactly one ace in two cards, choose one of four aces and one of 48 non-aces. That gives 192 unordered hands. Multiplying 4 by 51 would include two-ace hands and count each of those twice.

Constraints can also force several categories. If a toy deck has two cards of each of three ranks, a three-card hand with one of every rank has 2×2×2=8 possibilities. Order does not add a factor of six because the categories already distinguish the chosen slots.

Conditioning changes the denominator

An unconditional event is compared with all possible hands. If you are told the hand contains at least one ace, the denominator becomes the count of hands satisfying that information. For two-card hands, this includes one-ace hands and two-ace hands.

Uniformity within a constrained set needs justification. Conditioning on a fact about a uniformly dealt hand preserves equal weights among compatible hands. Conditioning on a player choosing to raise may not, because different compatible hands may trigger that action at different rates. That distinction leads into Bayesian inference.

Worked example

A uniformly dealt two-card hand is known to contain at least one ace. What is P(two aces given that information)?

Exactly one ace gives 4×48=192 hands.
Two aces gives C(4,2)=6 hands.
The compatible set has 192+6=198 hands.
The desired share is 6/198.

The conditional probability is 1/33, about 3.03%. Dividing by all 1326 hands would answer the unconditional question instead.

Common mistakes

Using “at least one” where the problem says “exactly one.”
Adding an ordering multiplier when category-labeled choices already identify a set.
Conditioning on an action as if every compatible hand had equal likelihood.

Keep these ideas

Turn exact constraints into disjoint category choices.
Check whether each physical set is counted once.
Restrict the denominator to the conditioning event.