🃏 Poker Hand Odds — Texas Holdem

An out is any unseen card remaining in the deck that would improve your hand to (most likely) the best hand at showdown. For example, if you hold four cards to a flush, any of the remaining nine cards of that suit is an out. Counting outs accurately is the foundation of every equity estimate, whether you use the rule of 2 and 4 or exact combinatorics.

It is a fast mental shortcut: multiply your outs by 4 to estimate your percentage chance of hitting by the river when two cards remain to come (on the flop), or multiply by 2 when only one card remains (on the turn). With 9 outs on the flop, for instance, the rule estimates roughly 36% equity. It sacrifices a little precision for speed, which is exactly why it is popular at the table.

Exact equity accounts for the true number of unseen cards. On the flop there are 47 unseen cards and two more community cards to come, so exact equity equals 1 minus the probability of missing on both the turn and the river: 1 − (nonOuts/47) × ((nonOuts−1)/46). On the turn there are 46 unseen cards and only one to come, so exact equity is simply outs/46. This avoids the small overestimation the rule of 2 and 4 introduces.

Monte Carlo simulation independently verifies the exact-equity formula by repeatedly drawing random unseen cards (thousands of trials) and checking how often an out actually appears among them. As the trial count grows, the simulated win rate converges toward the same value produced by the exact combinatorial formula, which is a useful sanity check and mirrors how real poker software estimates equity for more complex multi-way and range-vs-range scenarios where an exact closed-form formula is impractical.

Pot odds are the ratio of the call amount to the total pot after calling: call / (pot + call), expressed as a percentage. If your hand equity is higher than the pot odds percentage, calling has positive expected value (+EV); if equity is lower, folding is correct. The simulator's decision readout compares these two numbers directly and computes EV = equity × pot − (1 − equity) × call to show the actual expected profit or loss in dollars.