Understanding how to use odds

Many, if not most, recreational hold’em poker players really do not have a deep understanding of the math of chance and odds, and how best to use them.
We all realize poker is a game of chance – the likelihood of a particular occurrence. What’s the chance of a coin toss coming out heads-up? How often will it happen? With only two possible outcomes, the heads-up option will occur 1 out of 2 times. That’s the chance it will happen the next time you toss the coin. There’s no guarantee, but in the long run, you can bet on it.
It’s more complicated in a poker game. With a deck of 52 cards, there are so many different ways the cards can come up in your hand, and on the board in a hold’em game. The same applies to each of your opponents. To best explain, let’s examine a typical situation.
In a middle limit game, you connect with middle two-pair on the turn. However, there are many ways an opponent can end up with a better hand than your two-pair. In this case, there are three diamonds on the board. An opponent may already have a flush; or he might catch it on the river. (I hate to be rivered!)
Three of the cards are in sequence; so, an opponent could have a straight, which beats your two-pair, no matter how high. Then, too, there is the possibility an opponent has a set, or a higher two-pair than yours.
You have two active opponents vying for the pot. Each has lots of chips piled up in front of him. Player A, in an early position, is rather tight, but he just opened with a bet on the turn. He must have a strong hand, you contemplate.
Player B to his left, a rather aggressive player, raises the bet. You doubt he is bluffing, so he too must have a strong hand. With one more card to come – the river – what is the chance you will fill up for a full-boat? That would be nice, and undoubtedly give you the winning hand!
Let’s figure there are four cards (outs) that will fill up your two-pair, to yield the powerful full-house. You have seen six cards – your two hole cards and the four cards on the board. Thus, there are 46 (52 — 6) unseen cards that include your four outs. That’s four out of 46 chances you will fill up on the river – one out of 11.5.
If you were dealt 11.5 hypothetical hands in this situation, you could expect to make the full-house one time. Of course, that’s an average and applies in the long term; each case is independent of the previous ones.
We could express it as a percentage: 1/11.5 x 100 = 8.7 percent. Out of 100 hands in this same situation, you would expect to connect – on the average and in the long run – 8.7 times.
It is best to use the “odds.” What are the odds against making the full-house? If you fill up 8.7 times out of 100, then you miss 91.3 (100 — 8.7) times. So the “card odds” are 91.3/8.7 or 10.5-to-1 against making the full-house on the river.
How best to use info: Based on the way the betting has been going and your evaluation of your opponents, you are almost certain at least one of them has your two-pair beaten. At this point, the question is: Should I call the bet on the turn to see the river card, hoping to fill up?
Observe the pot; estimate the number of chips already in the pot. There are about 30 big bets. It will cost you two big bets to call. So, the pot odds are 30/2; that’s 15-to-1. The pot odds are higher than your card odds (10.5-to-1), so you have a Positive Expectation.
Make the bet. In the long run, it will be profitable for you.

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