Poker probability for quant interviews: outs, equity and pot odds

Poker probability questions test three skills that quant trading interviews care about: counting combinations without mistakes, updating a probability as cards are revealed, and comparing a probability with a price. You need only a few results by heart: 1,326 starting hands, 47 unseen cards after the flop, and the rule that a call is worth making when your equity exceeds the share of the final pot you must contribute.

Trading firms use card and dice problems because they have exact answers that can be checked quickly, and because the reasoning transfers directly to markets: estimate, observe, update, then decide whether the price is right. This guide works through the standard results, with every probability computed exactly. You can practise the updating step in the free Texas Hold'em odds trainer.

How many starting hands are there?

A Texas Hold'em player receives two cards from 52, so there are $\binom{52}{2} = 1{,}326$ possible starting hands. Most interview questions about starting hands reduce to counting a subset:

Event	Count	Probability
Any pocket pair	13 ranks × $\binom{4}{2}$ = 78	5.88%, about 1 in 17
Pocket aces	$\binom{4}{2}$ = 6	0.45%, 1 in 221
Suited cards	4 suits × $\binom{13}{2}$ = 312	23.5%
Ace–king, any suits	4 × 4 = 16	1.21%

A useful habit is to check a count from two directions. For suited hands: whatever the first card is, 12 of the remaining 51 cards share its suit, and 12/51 = 23.5%. Interviewers often ask for exactly this sort of second derivation.

What is an out, and how do you turn outs into a probability?

An out is an unseen card that improves your hand to one you expect to win with. After the flop you have seen five cards (your two and the three on the board), so 47 are unseen. Cards in opponents' hands are unknown to you, so they count as unseen too.

A flush draw has 9 outs: 13 cards of the suit minus the 4 you can see. The chance of hitting on the turn is

P(\text{turn}) = \frac{9}{47} \approx 19.1\%,

and the chance of hitting by the river is one minus the chance of missing twice:

P(\text{by river}) = 1 - \frac{38}{47}\cdot\frac{37}{46} \approx 35.0\%.

The same method gives the other common draws:

Open-ended straight draw, 8 outs: 31.5% by the river, 17.4% on the river alone.
Two overcards, 6 outs: 24.1% by the river.
Pocket pair that hits a set on the flop: 1 − C(48,3)/C(50,3) ≈ 11.8%, about 1 in 8.5.

The quick table-side approximation is the rule of 4 and 2: multiply outs by 4 with two cards to come and by 2 with one card to come. Nine outs gives 36% and 18%, close to the exact 35.0% and 19.1%.

Exact probability of hitting by the river against the rule of 4, for 1 to 15 outs

The rule drifts as outs increase because it ignores the chance of hitting on both cards. At 15 outs it says 60%, while the exact answer is 54.1%. Use it at the table; in an interview, give the exact expression first and the approximation second.

Outs are not equity

Hitting an out is not the same as winning. Some of your outs may also give an opponent a better hand: a card that completes your straight can complete their flush. And a hand can win without improving.

Equity is the share of the pot you would win on average if all remaining cards were dealt: your win probability plus half your tie probability in a heads-up pot. Against one random hand, before the flop, a simulation of 60,000 deals per hand gives:

Pre-flop pot equity against a random hand for seven starting hands

Pocket aces win about 85% of the time, but seven–two offsuit still takes about a third of the pot. Suitedness adds only about two points to ace–king. Note the qualifier "against a random hand". Against a player who raised, the opponent's range is far from random and every number here falls.

The Texas Hold'em trainer uses the same random-opponent assumption. It reveals the board one card at a time so you can estimate first, then compare. Before the river it simulates; on the river it enumerates all 990 opponent hands exactly.

How do pot odds turn probability into a decision?

A call is a trade: you pay a price for a share of a future payout. The pot holds $100 and your opponent bets$ 50. To continue you must call $50 to win a final pot of$ 200. Your required equity is

\frac{\text{call}}{\text{final pot}} = \frac{50}{200} = 25\%.

On the turn, with a flush draw and one card to come, you hit with probability 9/46 ≈ 19.6%, below the 25% you need. Ignoring future betting, calling loses money on average: 0.196 × 150 − 0.804 × 50 ≈ −10.8 dollars. Facing the same bet on the flop as an all-in, with two cards to come and no further betting, your 35.0% exceeds 25% and the call is profitable.

This is the same comparison a market maker makes between a fair value and a quoted price, and the same one a trader makes between implied and expected volatility. Interviewers listen for whether you state the break-even probability before computing yours.

Practice questions with answers

Probability of being dealt a pocket pair twice in a row? Hands are independent, so (78/1326)² ≈ 0.35%.
You hold two hearts and the flop has two hearts. Probability the board completes your flush by the river? 9 outs, two cards: 35.0%.
Probability that at least one of your two cards is an ace? 1 − C(48,2)/C(52,2) = 1 − 1128/1326 ≈ 14.9%.
The pot is $60 and you must call$ 20. What equity do you need? 20/(60 + 20 + 20) = 20%. The final pot includes your opponent's bet and your call.

For more practice in the same style, see conditional probability for quant interviews and expectation and waiting times. The free Bayes' rule trainer covers the updating step from a different angle.

Common mistakes

Counting 50 unseen cards after the flop. You have seen five, so 47 remain unknown to you, whatever the opponent holds.
Adding probabilities for two cards. 9/47 + 9/46 = 38.7% double-counts the case where both cards hit. Use the complement.
Using equity against a random hand after a raise. The opponent's range is conditional on their action. Updating that range is the real interview question.
Forgetting ties. Pot equity includes half the tie probability. On paired or straight-heavy boards, ties can be several percent.

References

The pre-flop equities are SelfMadeQuant's own Monte Carlo estimates: 60,000 heads-up deals per hand against one uniformly random opponent, with sampling error of about ±0.4 percentage points. All other probabilities are exact combinatorial calculations shown in the text.