Here's some useful background for anyone interested in poker. Frequently, when someone bets into you, and you are chasing a draw, you need to decide if you want to call or not. Below, we look at a couple ways to think about this.

Before we start, one important definition to get out of the way. The term equity/pot equity may be confusingly used to refer to a % of the pot or an actual $ amount. I will use %equity and $equity to differentiate between these. Both %equity and $equity look at what would happen on average if your hand was taken to show down without any further betting.

Before we start, one important definition to get out of the way. The term equity/pot equity may be confusingly used to refer to a % of the pot or an actual $ amount. I will use %equity and $equity to differentiate between these. Both %equity and $equity look at what would happen on average if your hand was taken to show down without any further betting.

Take an example on the turn where the pot is $100. then the opponent bets $25 into pot. We need to decide if we call or not.

We are doing the analysis below with the assumption that opponent is all-in (or that there is no more betting on the river). Intuitively, we might already think that it is cheap to call (we can call because the bet was small)- let's see what the math says:

**Simple Pot Odds**We are doing the analysis below with the assumption that opponent is all-in (or that there is no more betting on the river). Intuitively, we might already think that it is cheap to call (we can call because the bet was small)- let's see what the math says:

**Method 1 == pot odds required probability vs %equity**

Pot Odds

- the opponent just bet 1/4 of the pot.
- the pot odds that you are getting is 25:125 == 1:5
- ie. pot odds are post money (inclusive of bet)
- from the pot odds, the required equity (probability of winning) in order to call is 1/(5+1) = 1/6 = 17%

%Equity (== Your Hand's Actual Probability to Win)

- looking at your hand vs your opponent's range, what is your probability of winning?
- let's say you put your opponent on a pair or better and you just have a straight draw.
- you only win if u hit your draw (8 outs = 18% chance of hitting)

Now compare pot odds probability (17%) to your hand's probability to win (18%) and since hand probability > the probability required from pot odds, this is callable. (maybe in practice you might demand say a 5% buffer before saying it's callable?)

**Method 2 == EV calculation**

folding = $0

calling = 18% * 125 + 82% * (-25) = $2

so it's +EV to call. so call

Notice that here, you already take into acct the pot size (125) and the bet size (25).

Compare this to method 1 (the pot odds vs probability method) - the probability calculation doesn't take into acct the bet/pot ratio and that's why you need to compare pot odds to it in method 1.

I think in practice, method 1 is actually easier to work out over the board.

Compare this to method 1 (the pot odds vs probability method) - the probability calculation doesn't take into acct the bet/pot ratio and that's why you need to compare pot odds to it in method 1.

I think in practice, method 1 is actually easier to work out over the board.

## Implied Pot Odds

Now let's say we were not allin (there is more betting on the river). Let's say there's another $50 behind in effective stack size after the call. it's actually very easy to do implied pot odds

25:(125+50) == 1:7

25:(125+50) == 1:7

that's it. required equity is 12.5%.

no chg with hand equity. so 12.5% vs 18% == much bigger reason to call/you have much more juice

no chg with hand equity. so 12.5% vs 18% == much bigger reason to call/you have much more juice

Looking at the EV method, this is 18% * 175 + 82% * (-25) = $11

## Benefits of Offering the Correct Odds

On a side note, i think it is interesting to look at what the EV # means. One way to think about the expected value of your profit at each situation is your $equity - money put in. At each point of decision when you have to decide between raise/bet/call/check/fold, you are seeking to maximize your incremental profit.

Let's go through the scenario above where you and your opponent each put in $50 before the turn, and have $25 each left. At the beginning of the turn, your $equity is $18, so your accumulated profit since the start of the hand is $18 - $50 = -$32.

- if you could check it down (opponent hadn't bet) == you would get avg $18 from the pot of $100. Your incremental profit in this scenario is $0. Your accumulated profit since the start of the hand is still -$32.
- Opponent bets and you fold. You $equity dropped to $0 here from $18, and you also didn't put any more money in. So your incremental profit for choosing this option is -$18. Your accumulated profit is now -$32 - $18 = -$50.
- Opponent bets and you call $25 allin. You would get avg 18%*150 = $27 from the $150 pot. Your $equity increased from $18 to $27, but you also spent $25 calling. Your incremental profit = +$9 -$25 = -$16. Add this to your pre-calling accum profit of -$32 before to see that your post-calling accum profit is -$48.

Note that if you could, you would still much rather get option #1 than having to choose between #2 and #3. In option #3, your are choosing an action that has -EV (you lose another $16). However, choosing option #2 would have even worse consequences (-$18). All this is because your opponent had bet out at you when you had <50% in %equity. You either put in more chips being the underdog, or you fold- effectively losing your pot equity (the chance to draw out on the winner).

This has very interesting implications for when you are playing/betting optimally with the winning hand.

*By making an optimal bet, you win exactly your opponent's $equity since they should be apathetic to folding.*
Quick example to show this again: lets say your opponent still has 20% chance (one in five) to draw out on you in a $100 implied pre-money pot. Right now, your accum profit is 80 - 50 = $30. Optimally, you would lay 1:4 odds post-money, or 1/3 of the pre-money pot == $33. If we show that when you make this $33 bet, you are increasing your accum profit from $30 to $50, then we have shown a working example of the statement made above. It is obvious if opponent folds. If opponent calls, then your expected payoff is 80% * 166 = 133 and your cost is 50 + 33 = 83. Accum profit = 133 - 83 = 50.

Note that this optimal bet sizing is most important to get right when it comes to closer draws (ie. it matter less when you are 90/10 favorite already). I claim this because with a close draw (say 60/40), the opponent still has 40% equity in the pot, so betting correctly to win that 40% equity is likely to be hugely lucrative, vs winning the 10% equity is less so. This may be a reason for why we have more freedom to slow play with trips etc when we are already 90%+ favorites.

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