Chance to hit by the river
20.35%
5 outs, two cards to come
- Chance on the next card10.64%
- Odds against by the river3.91 : 1
- Rule of 4 estimate-0.35 points from the exact chance20%
The count behind 5 outs on the flop
Expect the draw to arrive in roughly 20 of every 100 hands. That supports a call against a bet of up to 34.32% of the pot.
- Unseen cards47
- Your outs5
- Cards that miss42
- Pairs of cards with no out861
- Pairs with at least one out220
- Chance by the river20.35%
- Chance on the next card10.64%
- Odds against hitting3.91 : 1
- Estimate from the rule of 420%
- Error of the rule of 4-0.35 points
Chance by the river = 1 - C(u - k, 2) / C(u, 2), with u = 47 unseen cardsFor 5 outs: 1 - C(42, 2) / C(47, 2) = 1 - 861 / 1,081 = 20.35%Next card = 5 / 47 = 10.64%
Of the 1,081 pairs that two cards can make from the 47 unseen, 861 contain only misses. The other 220 hold at least one out, which is 20.35% of the pairs.
Check the shortcut against the exact count and you find 20% versus 20.35%. A gap of 0.35 points is too small to change most decisions.
4 x 5 = 20% is all the shortcut asks of you with 5 outs on the flop. The adjusted rule is kept for flop counts above 8.
Which holding has 5 outs?
This count has no one textbook draw behind it. It is the total you get after adding up several draws and checking for overlap.
Not every out is clean. Here is the chance once shared outs are removed: from 20.35% down to 8.42% when 3 are shared.
| Shared outs | Effective outs | Chance by the river | Chance given up |
|---|---|---|---|
| 0 | 5 | 20.35% | 0.00 points |
| 1 | 4 | 16.47% | 3.89 points |
| 2 | 3 | 12.49% | 7.86 points |
| 3 | 2 | 8.42% | 11.93 points |
The bets 5 outs can call on the flop
Each row compares your 20.35% chance with the equity a call needs. You can call 2 of the 12 bet sizes with 5 outs on the flop.
Work the price backward from 20.35% and you find a limit of 34.32% of the pot: $34.32 into a $100 pot.
| Bet size | Equity a call needs | Margin over it | Verdict |
|---|---|---|---|
| 1/4 pot | 16.67% | +3.68 points | Call |
| 1/3 pot | 20.00% | +0.35 points | Call |
| 2/5 pot | 22.22% | -1.87 points | Fold |
| 1/2 pot | 25.00% | -4.65 points | Fold |
| 2/3 pot | 28.57% | -8.22 points | Fold |
| 3/4 pot | 30.00% | -9.65 points | Fold |
| full pot | 33.33% | -12.98 points | Fold |
| 1.25x pot | 35.71% | -15.36 points | Fold |
| 1.5x pot | 37.50% | -17.15 points | Fold |
| 2x pot | 40.00% | -19.65 points | Fold |
| 3x pot | 42.86% | -22.51 points | Fold |
| 4x pot | 44.44% | -24.09 points | Fold |
Nearby counts on both streets
Compare 5 outs with the counts on either side. Each step is worth near 3.79 points on the flop, and the other street is shown next to it.
- 5 outs on the turn: 10.87% on the river.
- 4 outs on the flop: 16.47% by the river.
- 6 outs on the flop: 24.14% by the river.
When a bet buys only one card
Facing a half-pot bet, compare 25.00% with 10.64% when you will pay again on the turn. A margin of -14.36 points says Fold, while the two-card view gave -4.65 points.
Counting the next card alone, you can call none of the 12 bet sizes, compared with 2 of the 12 bet sizes when you see both cards.
How this is computed
The chance to hit by the river from the flop is 1 - C(u - k, 2) / C(u, 2), where u = 47 unseen cards and k is your effective outs. For 5 outs that is 1 - 861 / 1,081 = 20.35%. The chance on the next card is k / 47, here 10.64%.
The rule of 4 figure is 4 x k, here 20%, and its error is that figure minus the exact chance in percentage points, here -0.35 points.
Assumptions: every unseen card is equally likely, and no card is removed for an opponent's hand, so this is the chance to hit your outs, not your equity against a named hand. A call is paid in full, so the pot after it holds the pot plus two bets. Here the pot is $100.
Computed from the closed form by the GeorgeLiPoker math library.
Updated
Questions
How good is a draw with 5 outs on the flop?
With 5 outs on the flop you hit by the river 20.35% of the time, about 20 of every 100 hands, and on the next card 10.64%. The odds against hitting by the river are 3.91 : 1, since 861 of 1,081 pairs of unseen cards hold no out.
What does the rule of 4 say for a count of 5?
The rule of 4 gives 20% for 5 outs against an exact 20.35%, an error of -0.35 points. That is close enough to use at the table.
What price can a draw with 5 outs on the flop afford?
At 20.35% by the river, you can call 2 of the 12 bet sizes. A half-pot bet needs 25.00%, a margin of -4.65 points, so against half the pot the verdict is Fold. The largest bet you can call is 34.32% of the pot, or $34.32 on a $100 pot.
How many points do I lose if outs are shared with a better hand?
Take them off the count first. With one shared out the chance by the river falls from 20.35% to 16.47%, and with 3 shared it falls to 8.42%, a loss of 11.93 points. The effective count is then 2 outs.
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