One card, then another
Multiplying probabilities across the turn and river, and why 'at least once' is easier backwards.
6 minute read
The turn and river are dealt one after the other, and the second draw depends on the first: once a card is gone, the deck is one smaller. Probabilities of sequences multiply, using the updated deck each time.
Both cards are hearts (backdoor flush)
10 ÷ 47 × 9 ÷ 46 = 4.16%
For draws you usually want the chance of hitting at least once in two cards. Counting the ways to hit once, or twice, gets messy. Counting the ways to miss both times is clean — and then subtract from one.
Hit at least once with 9 outs
1 − (38 ÷ 47) × (37 ÷ 46) = 34.97%
Miss the turn (38 of 47 cards), then miss the river (37 of 46).
The 15 highlighted cards are your outs.
What you’re drawing to
- Flush draw9 outs
- Backdoor straight draw0.93%
10 of the 1081 turn-and-river combinations make a straight.
- Two overcards6 outs
Pairing an overcard makes top pair, which is not always the best hand.
Shortcut vs. exact
| Cards to come | Rule of 2/4 | Exact | Gap |
|---|---|---|---|
| Turn only | 30% | 31.91% | −1.91 pts |
| Turn and river | 60% | 54.12% | +5.88 pts |
Quick check
You have 8 outs on the flop. Write the by-the-river probability as a product.
Show answer
1 − (39/47) × (38/46) = 31.45%.