Hi everyone

1. When calculating pot odds, we add the bet to the pot size in the denominator. For example, Hero bluffs $50 into $100. Hero's bet needs to succeed X% of the time, where X = Bet / Pot + Bet. In this case, X = $50 / $100 + $50.

Why is the calculation made in relation to the pot after Hero's bet? It seems weird to represent the pot size after the bet. I'm guessing it's because the money is considered dead once it goes into the pot, and when we win, we're winning that amount back. This feels counter-intuitive to me.

2. In Matthew Janda's book Applications of No-Limit Holdem, he gives an example where CO opens 3.5bb, and BTN 3bets to 11bb. In this instance, he says a successful 4bet wins 12.5bb total: 11bb from the BTN 3bet + 1.5bb from the blinds.

At first I was confused, because I was calculating the money won after the initial raise. To explain more clearly, I would calculate the total winnings to be 16bb. 11bb from the BTN 3bet, 1.5bb from the blinds, and 3.5bb from the our initial CO raise. This seems to fall in line with the logic of question #1 in this post, which treats money already committed as dead money. The way I can wrap my head around Janda's calculation is that we start the hand with 100bb, and if the BTN folds, we end the hand with 112.5bb, for a net gain of 12.5bb. I want to verify my thinking is correct, and to understand why these two problems differ with regard to pot odds and dead money.

1) think of what happens when you make a pot bet as a bluff if you want to calculate it your way
2) think you're probably right but I've not read the book in question for the context

Thanks for the response!

I see, because if X = Bet / Pot, then X = 1 if we make a pot-sized bet. That approach assumes we're risking $100, and the pot we win is $100. In reality, we win the pot and get our money back.

Let’s do the EV math for a simple hero all-in bet situation that is called:

EV = Expected (profit- loss).

Pot before hero bet = 100. Bet = 50 Villain call = 50

If hero wins, his profit is the pot plus villain’s call = 100 + 50 =150

If hero loses, he loses the amount he bet = 50

EV = eq*150 - (1-eq)*50

Set EV = 0 to find the break-even equity and you get,

eq = 50/(150 + 50) = 25%

You can rationalize this as follows:

After hero and villain bet, the total pot is 200. Of that total, hero contributed 50 at the decision point (when he bet) so his ‘fair share’ of the total is 50/200.

Thanks for your response. I'm getting confused when change the example to calculating Hero's fair share if villain calls.

I'm trying to figure out what the $50 in the denominator represents in my original example. I'm confused whether it represents Hero's bet, which is returned to him if he wins the pot, or his opponent's bet if the bluff is called. To put it another way, in the expression X = $50 / $100 + $50 above, does the $50 in the denominator represent Hero's bet or his opponent's potential call?

Mathematically speaking:
X = $50 / $100 + $50
X = A / B + C

X = how often Hero's bluff needs to succeed to breakeven, assuming we lose every time we're called
A = the size of Hero's bet
B = the size of the pot at Hero's decision point
C = ? (I'm trying to understand if this variable represents Hero's bet or his opponent's call)

C is hero’s bet but that doesn’t mean we count hero’s last bet as a win amount.

Because both hero and villain each bet 50, there could be confusion as to which bet is included in a specific calculation, so I’ll change the situation a bit to avoid that. Assume a pot of 100 when villain bets 10, making the pot 110. Hero then raises to 50 so villain has to call 40.

Below are three ways to determine required equity for break-even:

The EV equation for hero when he raises to 50 and villain calls is:

EV = eq*(110 + 40) - (1-eq)*50

If hero wins, he wins 110+40 = 150 or if he loses, he loses 50.

(1) For hero break-even (set EV=0). Then, eq = 50/(110 + 40 + 50) = 25%

Pot odds for hero is Reward / Risk. The Reward is the pot after all bets excluding hero’s bet or 110+40 = 150. The Risk is his bet of 50. So pot odds are 150/50 = 3 to 1.

(2) Required equity = 1/(Pot Odds + 1) = 1 / (3 + 1) = ¼ = 25%.

(3) Required equity = Amount contributed to Final Pot at decision time / Final Pot = 50/200 = 25%

Thanks for clarifying. I appreciate you taking the time to write another post.

I think I understand, when I'm looking at the equation X = A / B + A, what I'm seeing is the ratio of Hero's bet to the pot. The A in the denominator isn't representing money we're winning, but it does represent part of the total pot size, which Hero's bet is in relation to. If we didn't add Hero's bet to the pot, the ratio wouldn't accurately represent how much of the pot is Hero is entitled to. Looking at the full expression after villain calls, we would see Hero's bet as a ratio to the pot, his bet, and villain's call. This correctly represents all the variables at play, and allow us to solve for a breakeven point after all bets and calls have been made.

It's possible it feels unintuitive because you are thinking in terms of profit/loss rather than amounts.

1) Let's say we start with $200, and we play the example you mentioned 3 times. We lose the first two and win the third.

Profit and Loss
1) -$50
2) -$50
3)+$100

We are breakeven

Considering the same scenario but looking at what we put in/receive from the pot
1)put in -$50, receive $0
2)put in -$50, receive $0
3)put in -$50, receive $150

Again we are break even, we put in $150 and at the end we took out $150.

This is exactly the same as the profit and loss, it's just another way of thinking about the same thing.

How do we know that losing 2 and winning 1 gets us to breakeven? The formula you gave in your post

X = Bet / (Pot + Bet) = 0.5/(1+0.5) = ~0.3333 (or one third)

2) I didn't look at the section of the book, so I don't have context, but again, it depends how you want to think about it.

If you want to think in terms of Profit and loss, as you point out "with 112.5bb, for a net gain of 12.5bb." That's one way to think about it.

If you want to think about the amounts you put in/take out, the pot is 16bb, You will have to put in 7.5bb to match, and then more on top to 4 bet.

What you will receive from the pot will be 16bb+your 4 bet size, and you would again use the formula to calculate break even
X = 4betSize / (pot + 4betSize) = 4betSize / (16bb + 4betSize)

If I make my 4betsize small at 16bb, I need to win ~50%. If I make it large at 32bb, I need to win at ~67%, and so on.

It’s simply a risk/reward calculation. The break even probability is always risk/(risk+reward). In your scenario hero risks $50 for a reward of $100. To break even hero must succeed at least 50/(100+50) = 1/3 of the time. A bit of thought and consideration of modifications of that scenario will convince you that we can’t just divide the bluff amount by the pot amount before the bluff. What if hero bluffed $200 into that $100 pot? The bluff/pot method would indicate that hero must win 200% of the time to break even, which is obviously absurd. The correct formula gives 200/(200+100) = 2/3, meaning villain must fold at least 2/3 of the time for hero to break even.

Looked at another way: in repeated plays hero must win an amount equal to the amount he loses. Let N be some large number, and the number of times hero tries to bluff. Let x be the bluff amount and y be the pre-bluff pot amount. Let p be the probability of the bluff succeeding. Hero wins Npy dollars on successful bluffs and loses Nx(1-p) dollars on unsuccessful ones. Therefore for breakeven Npy = Nx(1-p), or simplifying py=x - px. Rearranging py+px = x, p(y+x) = x, giving p = x/(x+y), the same formula as in my first paragraph, with x being risk and y reward.