# Implied odds and expected value

In this poker guide we will study pot odds under extreme conditions. If there is a sincere possibility that your opponent(s) may go all in at the river you may have implied odds to call them, even though you may not have the pot odds. We will go to use an example to show this.

## Preflop implied odds by the potential of hitting three of a kind

You are holding 88 a player from under the gun raises 3 times the big blind and another player reraises. You are sitting on the big blind, do you call 8 times the blind with only a draw towards a set? You are currently holding a stack of 100bb.

Chance of flopping a set or a full house: 12,5% or 7 to 1

Pot odds: 13,5 to 8.

You can’t call based on pot odds, but let’s estimate the expected value.

## Estimating expected value based on assumptions

Both opponents have stacks of over 100bb.

There is a significant chance that the first player will rereraise all in, probably three out of ten times, inwhich case your 88 should probably be folded. The other 70% you will see the flop with two players and a pot of 27,5bb.

Lets say 50% of the time when you hit your set, one of your opponents wil call your all in on the river  (for the rest of his stack of 91 BB for a total of 27,5+91+91bb=207,5bb) and you will be winning the hand 80% of the time in such case. The other 50% of the time you will win a pot of 55 blinds ( two times the potsize on the flop, because it is likely for both players to at least try a last stab at the pot)

30% of the time you loose 9 big blinds.

70% of the time you will see the flop, and 12,5% of the time you will hit your set inwhich case you will win 27,5+91+91(the rest of the stack of one of your opponents plus the rest of your own stack)=207,5  bigblinds 80% of the time.

63% of the time you will loose 9 big blinds after seeing the flop.

7% (0,7*0,125*0,8) of the time you win

93% you will lose 9bb, which gives 0,93*9=-8,37 expected value

3,5% of the time you will win a pot of 55bb   =>expected value: 0,035*55=+1,89bb

3,5% of the time you will get one of your opponents all in while having a set for a total of 207 bigblinds. => expected value: 0,035*207,5=+7,26BB

The total expected value when playing this hand is: 1,89+7,245-8,37= +0,817BB

As you can see playing this hand because of implied odds is a very marginal play with a lot of assumptions, however the expectancy is positive.

As a general rule the stacksizes of you and your opponents should be at least 14 times the amount of chips invested preflop, because that is enough to earn to justify calling with 7% equity  (100bb/7%=14,28bb). In this case you only had 12 times the preflop investment, but you had two opponents to make up for it and you where sitting on the bigblind, which led to  an investment of 8bb.

Another rule is that you want to have two other players in the pot, because your goal is to maximise the chance of getting payed off. Even when your opponents don’t have a hand, the chance of one of them bluffing is big enough to at least get a slightly bigger pot.

In case you play with suited connectors the implied odds are less. There is a problem with the fact that the flop is often not enough to hit the nuts, which gives a problem of paying your opponents on the flop and the turn in order to see the river card. The chance of getting a flushdraw is: 10.94% and the chance of getting an 8-out straightdraw on the flop is: 10,45%. Besides that you could hit two pair on the flop 5% of the time. The problem is that you will need to pay a lot more then you need to pay to have the chance of hitting your set on the flop. This means you want to see the flop as cheap as possible with as many players as possible to pay you off in case you hit the nuts. Calling preflop because of implied odds may be a mistake, unless your investment is like 1/28th of the stacksizes.

This estimation is based on the following:

You hit two pair on the flop 5% of the time, inwhich case you have 50-70% chance to win the hand vs 2 players depending on how the board looks like. Usually you will win 3,5% of the time with the two pair.

You get a flushdraw 10,94% of the time, which means in that case you have 36% chance of getting the flush on either the turn or the river. For a total chance of  0,1094*0,36=0,0393, or 3,93%.

You get an 8-out straightdraw 10,45% of the time, which means in that case you have like 32% chance of hitting the straight on either the turn or the river. The total chance of this happening is 3,34%.

In case of backdoor draws or inside straightdraws, they usually won’t be improving often enough to justify calling any bets on the flop.

When talking about preflop implied odds, the 3,5% winrate with two pair can be used, which gives two times smaller implied odds then the former calculations of hitting the set, or 2*14,28=28,56. So you need stacksizes to be 28,56 times the preflop investment.

Incase you hit the flush and or straightdraw, you will recalculate your implied odds. For instance, if you have 36% chance of the nuts, you will be fine by paying of expensive turn and rivers incase your opponent has a large stack compared to the investments. Otherwise you will just only look at the pot odds 2 to 1 on the flop and 4 to 1 on the turn.

So lets say the potsize on the flop is 500 and you hit your draw. How to estimate your implied odds?

Lets assume your opponent is going to bet the full pot on both the turn and river to make your draw expensive.

He will bet 500, you call 500 on the flop and then he bets 1500 on the turn and you call 1500 on the turn. The pot will be 4500 to see both cards. If you see both cards you win 36% of the time. So in this case you want to get your opponent all in for 2  times as much as you have payed since the flop, or at least 4000 more. But on the flop your opponent needs to have at least 6000 chips behind to have enough implied odds to keep calling him on both flop and turn, which is 12 times the potsize on the flop.