I understand this is the relative value of playing vs. winning, but how do I determine the values? Am I supposed to count the pot? Does the cost of playing include hands already played, or just the next draw?
What are Pot Odds?
Pot odds are the value of the pot (how much you stand to win) in comparison to the cost of you calling, and are most often used to evaluate the value of making the considered call.
Calculating Pot Odds
Pot odds are a ratio,
Pot Value : Call Cost
To convert this ratio to an equivalent percentage, divide the
Call Cost by the sum of the
Pot Value and
$10 / $10 + $50 = 16.7%
To convert a percentage back to a ratio, subtract the
Pot Odds Percentage from
100, and divide the remaining number by the
Pot Odds Percentage:
(100 - 16.7) / 16.7 = 4.98 -or- 5:1 odds
Using Pot Odds to Calculate Value of a Call
Example: If the pot is $50, and the cost to call is $10 your pot odds are 50:10 or 5:1 (16.7%) when reduced.
If the odds of you drawing the winning hand are greater than
5:1 (16.7%), there is positive value in calling the bet. If the odds are less than the pot odds ratio/percentage, there is a negative value.
Obviously, even if the expected value of calling is positive you may still lose the hand... but over extended sessions these value calculations will be true statistically.
The trick often happens when using pot odds early in a hand (ex. Flop or Turn in Texas Hold'em) where the pot odds and chance of winning the pot may change again... this brings up the concept of implied odds and can be read about on Wikipedia and all over.
Pot odds are a way of determining whether it's worthwhile to continue with a hand.
Suppose that the pot contains $10 at the river and your opponent bets $2. You think that there's a 20% chance that you have the best hand, and that if you raise your opponent will fold a worse hand or reraise a better one, so your options are to call or fold. What should you do?
It will cost you $2 to call the bet, to win a pot of $12 (the $10 already in it, plus the $2 your opponent just bet), which means you're getting pot odds of 12 to 2 (or 6 to 1). Since you expect to win 20% of the time, you only need odds of 5 to 1 to break even, so you should call the bet.
If this wasn't the river, you would also need to calculate implied odds: the amount you expect to be put into the pot on future hands (which could push you towards either calling or folding).
Let's take a concrete example, so it's a bit easier to understand.
You are playing with Alice and Bob. The blinds are 50/100. You are the big blind, Alice is under the gun, and Bob is the small blind.
You post 50, and Bob posts 100 blind. Alice calls 100. Bob raises to 400. Currently, the pot contains 550. To call, you would need to add 350 to your 50. If you call, the pot will be 900. Therefore, the pot odds are 350 to 900 - you pay 350 to get a chance to win 900. If you believe that your odds are better than 350/900 (38%), you should call - that is ignoring anything that happens after the call.
Pot odds are the ratio between the amount of money available to win, and the amount you have to call. You should be counting the amount in the pot as the hand progresses, since you will not be allowed to touch the pot, and it usually can't be counted for you, during the hand.
For instance, if the pot is $25, and someone makes a $5 bet, your pot odds are (25 + 5) to 5, or 6 to 1. This means you can win 6 dollars for every 1 dollar of your call.
Relating it to your decision, you would need to be able to win at least one time in 7 to make the call profitable.
Pot odds are the odds you get when you make a call after a bet (or a raise) and it concerns a single hand. It is totally independent of the hands already played.
Am I supposed to count the pot?
Yes. They are typically expressed in "... to one". For example, if the pot has $15 and someone bets $6 in it (typically not a strong enough bet but that is not the point), then now the pot has $21 in it and if you have to pay $6 to call, you're getting 3.5:1 on your call ($21 divided by $6 gives 3.5).
It's a convenient way to determine if your call makes sense compared to your hand, especially if your hand is a draw-hand (for example you're trying to hit a flush or a straight).
Note that the pot odds are also called "explicit odds" and it's not enough to only take into account these odds: you typically also want to take into account what are called the "implied odds" (but that is a topic for another question).
Pot odds are the ratio of the current size of the pot to the cost of a contemplated call.
They are very closely related to explicit odds (fully and clearly expressed odds on offer right now) and implicit odds (not expressly stated, pertaining to probable explicit odds in later scenarios).
Specifically, odds relate to costs. Explicit pot odds relate to how likely your hand will win now versus the cost, and implicit odds relate to how likely your hand will win later (considering outs for you and your opponent/s) in the future.
Getting 3:1 pot odds means we have to win 1 in every 4 times to break even. To give some intuition, if we win 3 times what we risk every cluster of 4 hands, our net gain looks like
(-1 -1 -1 +3) + (-1 +3 -1 -1) + (+3 -1 -1 -1) + ... = 0
Anytime we win more often, we are making money.
More formally, we can derive this directly from our familiar equity inequality. We require
px - b(1-x) > 0 to be true to justify a call, where
p is the size of the pot,
b is the size of the bet*,
x is the probability we hit our outs, and
1-x is the probability we miss. We interpret this as: we win
px when our outs hit and we lose
b(1-x) when we miss, and we want the net win to be positive. Simplifying, we have
px - b + bx > 0, then
x(p+b) > b, and finally
x > b/(p+b). That is, the probability of our outs hitting should be greater than the bet divided by the new pot size, or we should be getting pot odds of
(p+b)-b:b, which is just
p:b. In other words, saying that we have correct pot odds to call is equivalent to saying that calling is a positive expectation action.
* When future bets are considered, we use the term implied odds. Otherwise pot odds generally refers to immediate or expressed odds.