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If we raise, our opponent might call with a variety of hands and I want to know my equity against the made hands (straight, set, flush, two pair) and the bluffing range (flush draws, gut-shots).
— AK vs JhTc — 17% vs.82%
- our opponent has a flopped straight with the nut flush draw. We have 4 outs.
— AK vs QQ — 17% vs 77%
- our opponent has bottom set of queens. Neither hand has a flush draw. We have 4 outs. There is a ~6% chance of a tie due to running hearts.
— AK vs 3h2h — 17% vs 80%
- our opponent has the worst possible flush. We have 4 outs.
— AK vs KQ — 86% vs 8%
- our opponent has a weaker two-pair. They have only 2 outs to suck out. There is a ~6% chance of a tie due to running hearts.
— AK vs Jh2c — 57% vs 42%
- our opponent has the nut flush draw and a gut-shot. They have 12 outs to win. The 9 remaining hearts to hit the flush and the 3 remaining Tens to hit the gut-shot straight.
Expected Value
<div>
</div>Fold EV
= (% opponent folds)*($300)
Call EV
= (% opponent calls) * [(% hero win)($300 + $625) + (% hero lose)(-$625)]
Total EV
= Fold EV + Call EV
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To do an example, let’s make assumptions:
- our opponent will call the raise 25% of the time and fold 75% of the time
- If we win ~20% of the time, what is our expected value?
Fold EV
= (75%)($300)
= $225
Call EV
= (25%)[(20%)($925) + (80%)(-$625)]
= (0.25)[(0.20)(925) – (0.80)(625)]
= (0.25)[185 – 500]
= $-78.75
Total EV
= 225 – 78.75 = $146.25
For the price of calling $625 more, our opponent would need to call us more often (higher than 25%) to reduce our fold equity. We are making a lot of $$$ every time our raise forces a fold. To do this, our opponent will need to start calling with a slightly weaker range of hands (drawing hands).
Let’s adjust our assumptions:
- our opponent will now call 50% of the time and fold 50% of the time
- since they are drawing more often, I estimate that we win ~30% of the time and lose 70% of the time, what is our expected value?
Fold EV
= (50%)($300)
= $150
Call EV
= (50%)[(30%)($925) + (70%)(-$625)]
= (0.50)[(0.30)(925) – (0.70)(625)]
= (0.50)[277.5 – 437.5]
= -$58.75
Total EV
= $150 – $58.75 = $91.25
By calling more often, our opponent has changed the EV in their favor. Our expected value decreased from $146 to $91.
Now, let’s get fancy.
What % win is needed to break-even when our opponent calls 50% of the time?
Fold EV + Call EV = 0
Fold EV
= (0.50)(300)
Call EV
= (0.50)[(W)(925) + (L)(-625)] where L = 1 – W
= (0.50)[(W)(925) – (1 – W)(625)]
= (0.50)[(W)(925) – (625 – 625W)
= (0.50)(925W + 625W – 625)
= (0.50)(1550W – 625)
Total EV
(0.50)(300) + (0.50)(1550W – 625) = 0
150 + 775W – 312.5 = 0
775W = 162.5
W = 0.209677 or 20.9% win is needed
This means, if our opponent calls us 50% of the time, we only need to win 20.9% of the time to break-even.
- If we win more than 20.9%, our EV increases.
- If our opponent calls less often than 50%, our EV increases.
