I was reading Essential Poker Math by Alton Hardin and I came across an example hand that talks about implied odds. Here is the hand:

Hero: A 10

Flop: 5 K 8

Villains $40 into a $50 pot.

- Our pot odds are 31% ($90:$40 = 2.25:1)
- We have 3 Aces and 9 clubs to use as outs which gives us a total of 12 outs.
- Using the Rule of 2 and 4, our equity is roughly 24% (12outs x 2)

The author notes that since the Pot Odds(31%) is bigger than our Equity(24%), that we should fold. But depending on the villian, this call could be correct if we knew we could extract more money later on if we hit our hand.

My questions to you guys would be...

- How to figure out what the minimum we need to extract from the villian to make this a +EV play, assuming we hit our hand on the turn?

- Is there a quick way to figure this out while in the hand?

I tend to think about this kind of problem in terms of odds, which is easier for quick proportional reasoning (for me, anyway).

You're getting laid 2.25:1 as 72:24 ≈ 3.15:1 shot. You essentially need to make up that 3.15 – 2.25 = 0.9 deficit in your odds. Multiply that difference by the number represented by the 1 in your pot-odds ratio (the amount you have to call, $40), so 0.9 × $40 = $36. You need to win at least $36 more by showdown to break even on the $40 call.

All you're really doing is figuring out how much bigger the pot would need to be right now for you to be getting proper pot odds. If there were $36 more, you'd be getting $126 for your $40, or 126:40 = 3.15:1, the odds against making your hand.

Thats what I thought, but wanted to make sure. Appreciate the feedback!

If you’re more comfortable with a direct equation, here is it for determining the future amount you must win in implied winnings to break even after villain made a bet of Bet into a pot of Pot and you have equity of eq.

Implied Winnings = (1-eq)*Bet/eq-(Pot+Bet)

For the OP problem, Pot = 50, Bet = 40 and eq= 24%, so

Implied Winnings = 0.76*40/0.24 – 50 – 40 = 36.67

I am more comfortable with the %'s. I appreciate your response. The only problem is, since I play LIVE a lot more than online, that equation would be tough to do at the table. Especially during a hand. I will have to sit down and think about a quick way to do that in my head much quicker.

If I was to round the 0.76 to 0.80 and 0.24 to 0.20, then it would look like:

0.80*40 / 0.20 - 50 - 40

The only problem is, Im getting = 70. Thats a big difference. Any chance Im doing the calculation wrong? Trying to find an easy way to do it in my head.

i come at it a slightly different angle to Jimulacrum

if you need 3/1 odds for the $40 call to be EV+ then the pot needs to be at least 3*$40= $120. as the pot at the moment is $90 you need to ask yourself will villain(s) put at least another $30 in future bets.

Exact Implied Winnings = 0.76*40/0.24 – 50 – 40 = 36.67

The card odds (1-eq)/eq is 76/24 or about 3 to 1. Your approximation is 4 to 1, a big difference. If you use 3 to 1 you have

Approx Implied Winnings = 3*40 - 90 = 30

Since the ratio was actually a little more than 3 to 1 you should add a little bit to the answer, say about 6, giving you 36

Note: The above is exactly what quasar did (except for adding a bit)

Thanks Quasar and Statmanhal!

I figured that out last night. Since I need 3 to 1 call , I went up to 3 and multiplied it by 40 to get 120. Then did 120-90 = 30 but added a few too it since it's little more. So $35 is probably what I would do in a hand since it's pretty close.

Also, if I didn't hit the flush on the turn, I'd keep that $35 in the back of mind, for when the villian bets again.

This is exactly the kind of thought process you need at the table. Looking at the manual calculations here is good practice, but multiplying and dividing multi-digit decimals while in action is a tall order for most folks (and more precision than you really need).

LOL yea I agree! Again, thanks for the help.