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Good poker requires proficiency in two areas: The human element, and the math. Here at Elite Poker most of our work will focus on the human element for two reasons: First, it is the more subjective area, and second, the math part of the game is extremely easy to learn. Sure, you could devote yourself to the study of game theory at post-doctorate levels and apply that knowledge to your poker game, but you could also learn 95% of the math you’ll ever need at the table in just an hour. We’re going to do the latter.
Any time you are faced with a bet during the game, the “pot odds” are calculated like this:
The size of the current pot
The amount you’re being asked to call
“The size of the current pot” is the sum of all of the money currently in play during the hand, so whatever is in the middle of the table, plus any amounts that are currently being wagered by the opponents who have acted before you. An example:
It’s NL hold’em with blinds of $1/$2. Before the flop there are two callers (you are the second one), then the SB folds and the BB checks. Before the flop is dealt, the money out there is collected into a pile and becomes “the pot” (in this case, “the pot” would be $7, $2 from each of the two callers + $1 from the SB + $2 from the BB).
On the flop, the BB checks and the first preflop caller makes a bet of $5. You call, and so does the BB. There are now $15 more dollars added to the pot, making it $22.
The turn comes, the BB makes a bet of $15, and the first caller folds, it is now your turn. What pot odds are you getting on a call?
$22 in the pot + the $15 bet = $37
The $15 you’re being asked to call
So, the pot odds are 37/15, which reduces to roughly 2.5/1 (just a tad under 2.5, to be precise).
Why is this important? Let’s say that all you have at this point is a flush draw, and that if you were to make that flush on the river, it would definitely be the best hand out there. If you do not make the flush on the river, you believe there is no way you can win the hand: The guy in the BB definitely has a stronger hand than you, and you do not think you’ll be able to make him fold on the river with a big bet as a bluff. In this case, the pot odds are the only thing that matter in your decision:
You’re being asked to invest $15 right now, and the payoff will be a profit of $37. For that decision to be profitable, making it would have to return a win more frequently than once every 2.5 times. For argument’s sake, let’s say that flush draw were only going to hit once every ten times. In that case, you would lose your $15 nine times before finally winning once and getting a $37 profit, so you would lose $15*9=$135, then win $37, for a net of –$98 over ten trials. In that case, calling the $15 would be an awful decision, since every time you do so your “expected value” would be -$9.80.
In reality (thankfully) that flush draw is going to come through more than one in ten times: It’s actually going to hit just under 20% of the time, or one in five times, which is the same as saying 4/1.
Knowing that, and knowing that your pot odds are 2.5/1, we can determine with 100% certainty that folding is the correct option here. We will only win the $37 in the pot one time for every four times we lose the $15, so over the long run, calling the $15 in order to see that river card is not going to win us money (it’s actually going to lose us $4.60 every time we call, no matter what the river card is. Can you figure out how I got that number?)
And that’s it: Pot odds. You calculate them, then you calculate the odds of you winning the hand, and if the pot odds are longer than the win odds, playing is the right decision. If the win odds are longer (as in our example, where they were 4/1 compared to the pot’s 2.5/1), then folding is the right decision.
Of course, calculating the pot odds is useless if you can’t calculate the win odds, so check back here next week for the next addition to this section!
Unlike Pot Odds, “Win Odds” are not something whose calculation is right there in front of you while you’re playing. To truly calculate your chances of ending a hand with the strongest cards, you’d have to know exactly what everyone else is holding, which we never do. So here what we’re going to focus on is how we can calculate the odds not of “winning”, or of having the strongest cards, but of improving our cards on the turn and/or river.
For those of you who have forgotten middle school probability, the chance of a thing — any thing — happening is calculated as follows:
In the context of a deck of cards, the “total outcomes” would be the number of cards in the deck that you do not know the value of. The “favorable outcomes” would be the number of those cards that help you. Here’s an example:
You are dealt JTo and the flop comes K-8-9 rainbow (the K-8-9 are all different suits).
You are pretty sure (because you’ve read my other articles and are now very good at poker) your opponent has made a pair of kings, so a J or a T on the turn would not do you any good, but a Q or a 7 would obviously give you the stone cold nuts and, in the case of the Q, could possibly give your opponent two pair (if he’s holding KQ) and give you a chance to bust him.
What are the chances you make your straight on the turn?
Let’s do the math, starting with the “total outcomes”: There are 52 cards in the deck, and we know the whereabouts of 5 of them (the J and T we’re holding, and the K-8-9 on the board), leaving 47 cards we do not know the value of. This makes the denominator 47.
Now, the numerator, “favorable outcomes”: We need a Q or a 7 to make our straight, and we know there are 4 of each of them in the deck, so there are 8 cards out there that can help us. This makes the numerator 8.
So, the formula is 8/47, or roughly .17, which is the same as saying 17%.
There is a 17% chance you’ll make your straight on the turn.
Now, I don’t expect you to be coming to a number like .17 in your head, it would be asking far too much in a game that requires constant calculations. So, I’ll let you in on an easy trick:
At any time in a poker hand, it’s likely the denominator of your formula, the “total outcomes”, is going to be 46, 47, or 50.
- Before the flop, of course, you’ll only know the two cards you’re holding, leaving 50 unknowns.
- After the flop, there will be 47 unknowns (like in the example above).
- The last time you would need this equation — right before the river is dealt — you’ll probably only know your two cards + the 3 from the flop and the 1 from the turn, leaving 46 unknowns.
Other than when someone accidentally flips a card during a hand, one of those three numbers will always be your denominator. Notice how each of the numbers are very close to “50”? This is the key to coming up with something like “17%” very quickly.
Since a “% chance” is basically the number of times a thing will happen out of 100 trials (25% is 25/100, 12% is 12/100, etc., right?), all we have to do is take our equation and convert it so that the denominator is “100”, and whatever we have in the numerator would become our % chance. Luckily, with our 3 most likely denominators so close to 50, converting them to 100 is as simple as multiplying by 2!
And, since we’re multiplying our denominator by 2, all we would have to do is also multiply the numerator (the number of cards that help us) by 2, and our formula would be converted, giving us our % chance without having to do any complicated math!
Because those denominators are rarely exactly 50, though, the easiest way to make up for the slight difference is to multiply your numerator by 2, but then add 1-3 percent to the result before settling on a final number (adding only 1% when you have very few outs, but adding 2-3 when you have more outs, like in the case of an open-ended straight draw + a flush draw + two overcards whose pairs would help you, which would be 21 outs).
So, in the above example we established that we have 8 outs, so we multiply that by 2 to get 16, then add 1 because 8 is a pretty small number, to get to 17 which is, of course, the right answer: No doing 8/47 in your head necessary!
If you’re unclear about why we need to add 1-3% to our number after we multiply by 2, think of it this way: If we are multiplying “8 over 47” by 2, we’re getting “16 over 94“, not “16 over 100“. To accurately bump up 16/94 to X/100, we’d have to make that 16 just a tad bigger, so we add 1. When we have a lot more outs than 8, we add 2 or 3 to the total because it will get us slightly more into the ballpark.
Now that we know how to quickly figure our chances of improving with the next card dealt, how do we do the same when it’s after the flop and we want to know our chances of improving on the turn or the river? Easy.
The chances of a thing happening over two trials is simply the same as the chances of it happening once, only multiplied by two. In the example above, we’re looking at a 17% chance of making our hand when one card is dealt, so we’d have a 34% chance of making our hand when two cards are dealt. It’s just that easy.
To summarize, if you want to calculate the chances of your hand “hitting” on the next card, just count the number of cards in the deck that help you, multiply that number by 2, then add 1 to the result*. Want the chances of “hitting” on either of the next two cards? Multiply that first number by 2.
5 outs? 11% it comes on the turn, 22% on the turn OR river.
9 outs? 19% it’s coming next, 38% you get it by the end of the hand.
*if the number of cards is small, like <10. If the number of “outs” you have is 10-20, add 2. If it’s 20+, add 3.
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