Nine outs with two cards
With 47 unseen cards, the exact by-river probability is higher than one-card 9/47 but not simply two independent attempts.
Poker Draw Odds Calculator
Enter an outs count, unseen-card count, and one or two cards to come. The poker draw odds calculator uses combinations without replacement, then compares the exact hit probability with the familiar Rule of 2 or Rule of 4 estimate.
Convert outs into exact probabilities
The exact model samples unseen cards without replacement.
How the result is built
Why “By the River” Is Not Two Times the Turn Chance
The turn card is not put back into the deck. Count the miss path without replacement, then subtract it from one.
9 outs / 47
Turn hit19.15%
Turn miss → river hit15.82%
Turn miss → river miss65.03%
Next-card hit19.15%
At least one hit by river34.97%
Rule of 436%approximation
1 − ((U − O) / U) × ((U − O − 1) / (U − 1))
The exact complement calculation
It is easier to count misses and subtract from one. For two cards, the model compares combinations containing no outs with every two-card combination, so turn-river order is not double counted.
Odds against and percentages
A 20% hit rate has 80% misses, or 4 to 1 odds against. This ratio describes frequency, not a payout or offered betting price.
Where the shortcut drifts
Multiplying outs by two or four is mental arithmetic. Error changes with the outs count and can become material for large draws; the result panel reports the difference in percentage points.
Worked examples
Zero outs
The hit rate is zero and odds against are unbounded, which the calculator reports without dividing by zero.
Frequently asked questions
Should unseen cards be 47 on the flop?
Usually yes when your two cards and three board cards are known from a 52-card deck.
Are turn and river independent?
No. The turn card is not returned to the deck before the river.
What does odds against mean?
It is the expected misses per hit, expressed as x to 1.
Is the Rule of 4 exact?
No. It is an approximation shown beside the combination-based result.