Question of the interpretations of Pot Odds and Pot Equity, MATH

Question

In the middle is 10 dollars, someone bets 5. I have to call 5 dollars in order to win 10 dollars.

My Pot Odds - 5:10 = 1:2

I can interpret it as that i can lose 2 games and win 1 and I ll be break-even. Therefore I must win 1/(1+2) of the time (which is 33 and 1/3 %).

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And I came with another inrepretationan and I want to ask if it´s correct.

If I have a similar situation with Pot Odds - 2:3, which literally means that I can lose 1 and a half time if I win once, meaning I need to win 1/2.5=40 % of the time

The other way to look at is the amount of money that was given into the pot by myself... which is in case of pot odds 2:3... 2/5=40 % of the pot. So the 40 % of they money in the pot is my money I gave there.

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QUESTION 1) In this interpretation, if I have so good cards that I have the chance of winning ALSO 40 % of the time, does that mean that I win on average 40 % of the money in the pot and therefore i am break-even? Because I (on average) win 40 % of the money in the pot and 40 % of the money in the pot is given by me? So it returns back?

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QUESTION 2) Does money in the pot * chance of winning mean the amount I win on average (correct me or say if it´s right what I say in the last paragraph)?

formula:

money * probability = 10 dollars * 0,4 = 4 dollars

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QUESTION 3) Interpretations of the given formula

FIRST INTERPRETATION: If yes, why does that work? Only because it can be interpreted as that I win 4 times 40 dollars and the average I won during 10 games was 4 dollars per game (10*4 and divede by 10)?

SECOND INTERPRETATION: I think the second interpretation of why this formula works is that it can be seen as weightened average.

(10 dollars * 0,4 + 0 dollars * 0,6) / 1

0 dollars, because in 6 out of 10 you "win/lose, doesnt really matter how u call it" 0 dollars, the other 4 out of 10 times you win 10 dollars.

And therefore we can look at the problem as the weightened average of the amount of 10 dollars which has weight 0.4 and number 0 which has weight 0.6. I am not sure about this one, i just made it up, I am not sure about this one and I´d like to hear if it´s correct and if yes, why. Because i am not 100% sure of the concept of weights in averages.

Answer 1

[tl:dr, go to last couple paragraphs]

I'm having trouble fully understanding your question in the way you put it, but I'll try to address what I think are the issues that you're having trouble with.

First of all, pot odds and odds of winning are both ratios and thus comparable to each other, and indeed they should be compared to each other. For example, (staying with the 40% figure you use) let's say you're facing a bet of $40 into what had been a $20 dollar pot--so you need to call $40 to contest what is now a $60 pot. Your pot odds are then the amount in the pot ($60) to the amount you have to call ($40), so you're getting 60:40, or 3:2 pot odds.

You would compare this pot odds ratio to the ratio of your probability of losing versus your probability of winning. For example, if you expect to win 40% of the time and lose 60%, then that gives a ratio of 60:40 and is breakeven with the pot odds. You want the pot odds to be favorable, thus bigger, than your ratio of losing:winning probabilities. So if you expect to win 45% of the time, the new ratio is 55:45 and it will be profitable for you to call.

In practice, the hard part about this is scaling the numbers so that they are comparable. That may come through practice with working through examples and precise calculations, but in real time most people are safe to use some rough estimations with common fractions.

Equity is measured in dollars (or chips or whatever currency) and can thus be compared to a similarly measured number, namely the amount you're facing to call. Using the same example as above, say that after you've made a call, the final pot is $100. You believe your probability of winning is 40%. Then your probability of winning (40%) can be multiplied by the whole pot ($100) to get your equity, which will be $40. [Yes, this is the same as ((40% * the pot of $60) + (40% of your call of $40))--simple distributive property--the end result is the same no matter how the money got into the final pot]

What this number, equity, can be compared to is the amount you're facing to call. In the example where you're facing a bet of $40 then you're looking at a breakeven situation where your expectation is $0. Your expectation, or Expected Value (EV) is the average profit or loss you may encounter. It is also measured in dollars and you want it to be positive. In this situation, if your chance of winning is 45% then your equity will be $45. Compare that number to the amount you'd have to put in ($45 - $40) and you end up with a positive expectation of $5. In other words, you'll win $5 dollars on average for this hand.

Yes, one way of thinking about this is that (going back to the 40% chance of winning) if you played out this hand 10 times you'd expect to win 4 times with a profit of $60 and lose 6 times with a loss of $40 for a net of $0. With a 45% chance of winning, you could say you'll win 4.5/10 at a gain of $60 and 5.5/10 at a loss of $40, which amounts to +$50 over 10 hands, for an average gain of $5 which equals the EV from above. It's more practical though to just use straight percentages and calculate out a single hand than to add a denominator just to reduce by it at the end.

The way this works out to the same thing is thus:

W = your chance of winning (%)
L = your chance of losing (%)
D = final amount in pot ($)
C = amount you have to call ($)
P = amount already in pot, before call ($)

your expectation based on equity is then (WD - C) = (W(C + P) - C) = (WC + WP - C) = (WP + (1-L)C - C) = (WP - LC) which is another way of calculating the expectation as in the previous paragraph. We're interested if WP > LC. Do a little more algebra, so that P/C > L/W and now we have two ratios to compare that translate into pot odds and odds of winning and viola!, you have the calculation that was done in the second paragraph.

Hopefully that helps you see the connection of how all this is related. It's really just different ways of looking at the same thing, but some things are in $, some are %, some are ratios--you just have to be sure that you're comparing like things. Sometimes you may want to know that you expect to earn $20 on average by making this call or else you may want to know what % of equity you need at a minimum to be making a call. The way you calculate things out depends on what question you're really trying to answer.

Answer 2

What is in the middle is called the pot
If someone bets $5 into a $10 pot the pot is now $15
So facing the bet I am getting $15/$5 pot odds = 3:1
I am risking $5 to win $15

What people can miss on pot odds is if you win you get your bet back
So if I call 4 times and win 1 I am even
-5$
-5$
-5$
-5$ +5$ +$15
Net = 0

If you are getting 3:1 then you only need to win 1/4 of the time

If you have cards that have better than 3:1 odds then calling is profitable over the long haul. Even facing unpaired higher cards you are better than 3:1 and should call.

For pot odds at the turn see this

Weights comes into the more complex Expected Value (EV) where for example you give a weight to your opponent will fold. But for just deciding immediate pot odds to call the calculation is pretty simple.

: and / (%) are just different ways of looking at the same thing
As long as you are consistent the calculation should bet the same
A lot of players use : as for me the math is easier for me to run at the table and easier to memorize (bet + pot) / bet >= (losing outs) / (winning outs)
Remember it as you are on the bottom - your bet and your outs
On the turn there are 46 cards out and if on a flush draw there 9 outs
If they bet is the pot then
(bet + pot) / pot = 2 ? >= (46 - 9) / 9
2 ? >= 38 / 9
That is a clear no - need to be getting 4:1 to make that call (based on immediate odds)
If you get some weird numbers then you can do this
2 * 9 ! >= 38
18 ? >= 38 NO

From the other side if I am on a draw I calculate exactly how big a bet I can call before the bet is made. And I can do that with : a lot faster. I have all the major draws memorized. If you are getting money or near money then you want to call like you have a made hand and you should have been representing a made hand. If you hit you don't want to give away you were on a draw as a good player will shut down.

I see no value on faking a draw on a made hand. I think trying to give off false tails only makes your standard play more obvious.