How this instrument works
In poker, an 'out' is any remaining card that improves your hand into a winner — nine outs for a flush draw, eight for an open-ended straight draw, four for the last card completing quads. Most players estimate their odds of hitting an out using the quick 'rule of 4 and 2': multiply your outs by 4 with two cards to come, or by 2 with just one card to come. It's a fast mental shortcut, but it's an approximation, not the exact figure.
This instrument calculates the true probability using the hypergeometric distribution — the correct way to compute odds when drawing cards without replacement from a fixed deck. It first finds the probability of missing every one of your outs across all the remaining draws, using the ratio of ways to pick only non-out cards versus all possible draws, then subtracts that from 1 to get the probability of hitting at least one out.
The exact math and the rule-of-4-and-2 shortcut agree closely for small numbers of outs and cards to come, which is why the shortcut is popular at the table — but they diverge more as outs or draws increase, which is exactly when knowing the precise figure matters most for a big pot decision.
- Enter Cards remaining in the deck — 52 for a full unseen deck, fewer if you're accounting for cards you can already see.
- Enter Outs (cards that complete your hand) — the count of remaining cards that turn your hand into a winner.
- Enter Cards still to come — 2 for turn and river combined, 1 for the river only after the turn has already fallen.
- Read Probability of hitting at least one out for your exact chance of completing the draw across the remaining cards.
- Probability of missing every out shows the complementary figure — useful for comparing against pot odds when deciding whether a call is justified.
Worked example — a 9-out flush draw with two cards to come
Enter 52 for Cards remaining in the deck, 9 for Outs, and 2 for Cards still to come — the classic flush-draw-on-the-flop scenario. The instrument computes P(miss) = C(43, 2) / C(52, 2) = 903 / 1326 = 0.680995.
Probability of hitting at least one out = 1 - 0.680995 = 0.319005, or about 31.9%. That lines up closely with the well-known poker 'rule of 4' shortcut (9 outs x 4 = 36%), which is in the right ballpark but noticeably higher than the exact figure — a gap that matters more the larger your out count gets.
Questions
What's the difference between this and the rule of 4 and 2?
The rule of 4 and 2 is a fast mental-math approximation: multiply your outs by 4 with two cards to come, or by 2 with one card to come. This instrument computes the exact hypergeometric probability instead, which is more accurate, especially with larger out counts where the rule of 4 in particular starts to overestimate your real chances by a noticeable margin.
How many outs does a flush draw or straight draw have?
A flush draw (four cards of a suit, needing a fifth) has 9 outs — the remaining cards of that suit. An open-ended straight draw (four cards in a row, needing either end) has 8 outs. A gutshot straight draw (needing one specific rank in the middle) has only 4 outs. These are standard counts assuming a full, otherwise unseen deck.
Should I use 1 or 2 for 'cards still to come'?
Use 2 if you're deciding on the flop and want your combined odds across both the turn and river. Use 1 if you're already past the turn and only the river remains, or if you specifically want your odds of hitting on just the very next card rather than across multiple remaining cards.
Why does the deck size matter if I'm not seeing every card?
Cards remaining should reflect every card you can't currently see — the full 52 minus your hole cards, the board, and any cards you know are dead or discarded (in games where that applies). In standard Texas Hold'em with just your two hole cards and the board visible, most players simplify and use 52 minus known cards, which this instrument's default assumes unless you adjust it.
Is the hypergeometric distribution the right model for poker odds?
Yes — it's the standard distribution for probability problems that involve drawing cards without replacement from a finite deck with a known number of 'successes' (your outs) and 'failures' (everything else), which is exactly poker's situation. It's the same math used for problems like drawing colored balls from an urn without putting them back.