The Implied Odds Formula, With Algebra

Deep stacks are a reason to peel a draw, but “deep enough” is a number, not a feeling. The implied odds formula produces that number. If you can realistically extract X on later streets, the call breaks even; if you can’t, it’s a leak wearing the costume of optimism.

Deriving it

Break-even means the money you win when you hit, weighted by how often you hit, equals the money you lose when you miss. Let p be your hit chance, C your call, P the current pot, and X the extra chips you win on later streets.

When you hit, you take everything in the middle at showdown: P + C + X. When you miss, you lose your call, C. Setting the expected outcomes equal:

p × (P + C + X) = (1 − p) × C + p × C

The right side collapses to just C, because you risk the call whether you hit or miss. Solving for the pot you need at showdown:

P + C + X = C / p

And so the extra beyond what’s already there:

X = (C / p) − P − C

That’s it. The formula answers the only question a draw poses: how much more do I need to win when I get there?

Working a gutshot

You hold 6♠ 5♠ on K♦ 8♥ 4♣. A 7 makes the 4-5-6-7-8 straight and nothing else helps — a clean gutshot, 4 outs. On the turn your one-card chance is 4 / 47 = 8.5%, so p = 0.085.

Villain bets, making the pot $40, and asks you to call $20.

• C / p = 20 / 0.085 = 235
• X = 235 − 40 − 20 = $175

Straight pot odds here are 2-to-1, which a gutshot fails badly. But the formula says: if the stacks are deep enough to win an extra ~$175 on later streets when you hit, the call breaks even. With $300 behind and a straight nobody puts you on, that’s realistic — exactly the spot where implied odds rescue a call raw pot odds reject.

A reference table

All rows use a $20 call into a $40 pot, pricing one card:

Draw (one card)	Outs	p	C / p	Extra X needed
Gutshot	4	8.5%	$235	~$175
Open-ended straight	8	17.0%	$118	~$58
Flush draw (river)	9	19.6%	$102	~$42
Flush + straight combo	15	31.9%	$63	~$3

The pattern is clean: fewer outs demand a bigger future payoff. A 15-out monster barely needs implied odds; a gutshot needs a small fortune waiting behind.

The stack-multiple shortcut

There’s a faster mental version. The call is break-even when the showdown pot reaches C / p. Divide by the call and the pot you need is a multiple of it:

total pot needed ÷ call = 1 / p

• Gutshot, p = 0.085 → the final pot must be about 12× your call.
• Open-ender, p = 0.17 → about 6×.
• Flush draw on the river, p = 0.196 → about 5×.

Some of that multiple is already in the pot; the rest has to come from later betting. If the effective stack can’t physically hold the multiple your draw demands, the call loses no matter how the hand runs out. It’s the same shape as the set-mining rule of thumb — rare hits need deep stacks, common hits don’t.

Where the inputs betray you

The algebra is exact; the numbers you feed it aren’t. Three ways X flatters your edge:

• Reverse implied odds. Hitting your straight into a made flush wins less — or loses more — than the formula assumes. Discount X whenever your “winner” can still be second best.
• You won’t get paid. The formula bakes in extracting X, but a scary board or a cautious opponent can shut the action off the moment you hit. If the money won’t come, don’t count it.
• Dirty outs. Count your outs honestly first. Two outs that also complete a bigger draw for your opponent inflate p, shrink the X you think you need, and bury a losing call under tidy arithmetic.

Memorize the one line — X = (C / p) − P − C — and it converts a vague “deep enough, worth a peel” into a threshold you either clear or you don’t. Pair it with the pot odds baseline and the rest of the odds and math hub.