Using poker math concepts can help to win hands

Understanding and using poker math concepts can easily be the difference between winning or losing. You need not be mathematically inclined to do so. Today, let’s focus on (1) probability (often called “chance”) and (2) odds.
Probability: It’s an essential element in the game of poker. Don’t let it scare you. According to Wolfram MathWorld, “probability…studies the possible outcomes of given events together with the outcomes’ relative likelihoods and distributions.” It’s the chance that a particular event will occur, often expressed as a fraction or a percentage between 0% (zero probability) and 100% (certain to happen).
Flip a coin. Over the long term, it will land heads-up one out of every two tosses. The probability of that event is 1/2 or 50%. Playing hold’em, your first hole card is an ace. What is the probability (chance) that the second card also will be an ace – giving you pocket aces, the best possible starting hand. Easy: There are three aces still in the deck out of the remaining 51 cards.
So the probability of catching a second ace is 3 out of 51; that’s 3/51 = 1/17 = 5.8%. Interpretation: When your first hole card is an Ace, you can expect to get a second ace 5.8 out of 100 times – quite rare! That is the probability or chance that event will occur.
Odds: More often, we use the card odds which are estimated from probabilities. Starting with one Ace, what are the odds of your next hole card being another Ace? That’s simply the ratio of the probability (chance) of being dealt a second ace in the hole compared to the probability (chance) you will not.
As noted above, you have three chances to catch the second ace out of the 51 remaining cards in the deck, and 48 (51-3) chances of missing. The card odds, therefore, are 48-to-3, or 16-to-1 against it.
Types and uses: Poker players are concerned with two types of odds – Card and Pot. With a drawing hand (as is often the case), what are the odds of catching one of the cards you need to “make” your hand – presumably the winning hand? Example: Starting with two high clubs in the hole, the flop brings two more.
Now, holding four-to-a-club-flush, what are the card odds against completing the flush? Answer: The deck (unseen cards) includes nine cards of “your suit” (13 less the four you “have”). These nine cards are your “outs.” And, there are 52-5 = 47 unseen cards in the deck. (You have seen your two hole cards and the three cards flopped – 5 in all.)
Since nine of these unseen cards are clubs, then the rest, 47 — 9 = 38 cards, will not help you. Therefore, the card odds are 38-to-9 against you. That’s 4.2-to-1 against catching the club flush on the next card – the turn. We can round off the card odds to 4-to-1 against.
Compare these card odds with the pot odds – the amount of money (chips) in the pot vs. your call bet to see the turn. The pot contains $12, including your opponent’s $4 bet. To call the bet in order to see the next card, your pot odds are $12-to-$4 = 3-to-1.
Meaning: With 4-to-1 card odds against you, compared to the 3-to-1 pot odds, in the long run, you would expect to lose more than you win. On this basis, your call would be a Negative Expectation bet. But wait a minute! Don’t fold!
When the card and pot odds are close, consider the Implied Pot Odds – how many more chips will be added by your opponents during the rest of this hand. With two or more opponents staying, there is bound to be considerably more of your opponents’ chips going into the pot on the turn and river.
When you compare the estimated Implied Pot Odds to the present card odds, they are much higher. You have a Positive Expectation and your call is a wise investment.
We welcome your questions about the math of poker.

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