How do I calculate the odds of making my hand for any given game of poker?
The simple case, for making your hand on the next turn, is to add up the number of cards that can give you the winning hand on the next turn, and divide by the total number of unseen cards left in the deck. This means that, in games like Stud, you need to remove any potential outs that have been folded by others from the cards that can give you a winning hand.
In general, if you have seen 5 cards, and you have 4 to a flush, there are 47 unseen cards, and 9 that will give you a flush. If you think a flush is the winning hand, you have a 19.14% chance. If you are playing a flop game and you are up against a set, you have to discount the one flush card that would pair the board, which means you have 8 outs out of 47, or 17%.
So, for the "next card" solution, you have:
When there are multiple cards left, more advanced statistics become involved. You have to consider cards that may give your opponents the best hand, as well as conditional probabilities. In flop games with only the turn and river to go, and shared board cards, an approximation of the formula would be:
In that case, opponent outs are outs that will give your opponent a better hand than you. Depending on the hand you are drawing to, and your opponent's hand, the formula can be simpler - in the case of a flush draw against a set, you have 8 outs twice, but your opponent will have 6 cards on the turn that give him a full house, and 9 cards on the river that will do the same.
A good rule of thumb I always use for calculating odds is multiply number of outs(13 with 4 on the flop and 2 on the turn. This way it will be much easier to remember and you are never that far away from the correct percentage.
If you do not understand the term "outs". This is basically just the number of cards you need to hit your hand. If you are holding for instance 2 spades in your hole and there are 2 more spades on the flop, you are holding a so called flush draw. There are 4 spades showing now and with a total of 13 spades in the deck, this leaves 9 more spades possible to hit on the turn or river. 9 outs for hitting your flush.
If you wanna use my easy to remember rule of thumb this makes it:
On the flop = number of flush outs x 4 = (13-4)x 4 = 36%
On the turn = number of flush outs x 2 = (13-4)x 2 = 18%
These numbers are somewhat higher than the absolute correct percentage though. Just remember to subtract a couple of percentages after calculating and you are never far away from the correct one :)
By the way,
You can use the rule of 2/4 as described above, or by thinking in reverse, do some simple math to get the odds of completing your hand by the river.
Instead of directly calculating the odds of building your hand, think of the odds of NOT getting your card -- it makes it much simpler. As an example, after the flop you have 4 to a flush. What are the odds of NOT getting your flush with 2 cards to go? There are 9 cards that will fill the flush. 47 cards remain. So on the turn, your odds of NOT hitting are 38/47, and if you are unsuccessful on the turn, then your odds on the river of not hitting the suit decrease (slightly) to 37/46. Then the probability of not hitting on either is simply the product of the two odds: 38/47 * 37/46 (and conversely, your odds of hitting are 1 - [38/47 * 37/46].
If you can do long division in your head, you would know that 38/47=~.81, and 37/46=~.80. .8 x .81 =~.65. So you have a 65% chance of NOT hitting the flush, and a 1-.65 = 35% chance of hitting it.
Since long division in your head may not be a skill you wish to develop, just use approximations -- you will be close enough to determine your proper betting behavior. Reduce the fractions to see that 38/47 and 37/46 are both around 4/5 or .8, with a shade left over (9*4 = 36, 9*5 = 45, and 36/45 is very close to the division you want, but slightly less than either 38/47 or 37/46). multiple them together you get .64. You know the real answer is a shade over 64%... which gets you very close to the actual answer without doing complex mental gymnastics.
The harder part is calculating your opponents odds. The same technique applies, but you must make informed guesses as to what he is holding. Building a profile of your opponents' hands is the essence of Texas Hold 'em decision analysis, but not germane to the calculation question.
Suffice to say that you should be very leery of playing with idiot-savants who can do Rainman-like calculations rapidly in their heads... they are going to be a step ahead in knowing the odds of beating you. You can get better and faster with practice, though!
It's hard to do all the exact math to get the exact percentages while you're sitting at the table and everyone is staring at you while you noodle your chances. You have to do it all in your head in less than a minute. The other methods described here are actually more accurate, but I don't think that getting the exact percentage is that important, but you need to be close.
The way I do it on the fly is:
Some basic out counts after the flop are:
Of course, there is always the dreaded redraw which actually reduces these percentages, but you can include this easily enough in your calculations accordingly. And a runner-runner is usually less than 2%. Yup, a runner-runner is a 1 in 50 shot. Damn 2% suckout river-rats.
Your action, once you've calculated your odds, depends on your pot odds and perhaps implied pot odds. If your pot odds are better than your card odds, then you have a positive Expected Value (EV). If not, then you have a negative EV - so just fold.
Suppose you're at the flop with a pair and you want to know your chances of improving to a set by the end of the hand.
You've seen five cards from the deck (the two in your hand and the three on the flop); this means there are 47 cards in the deck that you haven't seen, and two of them will help you. Thus, the turn has a 2/47 chance of improving you to a set. If you don't improve on the turn then you still have two outs, but there are only 46 cards left in the deck, so you have a 2/46 (or 1/23) chance of improving on the river.
The odds of making your hand are equal to one (100%) minus the odds of not making your hand; in this case, that would be a 45/47 chance of not improving on the turn and a 44/46 chance of not improving on the river; this is a combined (45/47)*(44/46)=~91.58% chance of not improving on the turn or the river, which means you have approximately an 8.4% chance of hitting your set.
(Of course, this doesn't always mean an 8.4% chance of improving; you could unexpectedly backdoor a flush or straight)
The brute force method is to take all of the cards in the deck, deal out every possible combination of cards for a single player in the full course of the game and mark the total amount of times each possible hand occurs. Once you have the full stats, you can work out percentages for each hand by adding all of the hand counts together and then dividing the amount for each type of hand by the total.
Assume for a second there is a game with a deck of 7 cards, and each player gets 2 cards. The cards are a 2, 2, 4, 4, 6, 6 and 8. The hand ranks are that you either have a high card or you have a pair.