Poker players use odds to figure card’s worth

With a drawing hand, well-skilled poker players use the card odds along with the pot odds to decide if paying to see the next card is worth the investment. But there are times when they may make a costly mistake. Let’s start by defining our terms.
Pot odds: The amount of chips (money) in the pot divided by the cost to stay in the hand and not fold to a bet. Example: After the flop, there are 48 chips in the pot. You will have to bet (invest) four chips to see the turn. Then you are getting 48/4 = 12-to-1 pot odds on that bet.
Card odds: (sometimes called the “drawing odds”). The odds against catching one of the cards you need to make your hand – hopefully the winning hand. Then, you wisely call the bet if the pot odds are higher than the card odds – a Positive Expectation. For years, skilled players have been estimating the card odds. On occasion, they may have erred unwittingly to a significant extent – myself included. That error is bound to have cost us money.
Current Method: Let’s illustrate how the card odds are usually figured today: Suppose you are on the button and have four cards to an inside straight on the flop with A-2 suited in the hole, and 4-5 offsuit on the board. (There is little chance anyone is drawing to a flush.) The card odds are estimated by multiplying the number of outs by 2 to get the probability of making your hand on the next card (the turn, in this case).
Most skilled players regard this hand as having 4 outs – the four unseen treys in the deck. So, 2 x 4 (outs) = 8 percent probability of making the 5-high straight. Then, the probability of missing is 100 — 8 = 92 percent. The card odds are simply the ratio of misses to hits, 92/8 = 11.5-to-1 against connecting.
Since the pot odds (in our example hand, above) of 12-to-1, are higher than the card odds at 11.5-to-1, calling to see the turn gives you a Positive Expectation. You will make a profit in the long run. The problem is you are giving yourself more outs than is realistic – more than you are really entitled to.
Be More Realistic: Let’s think this through. Can you really expect all four treys to be available in the deck? We are at a full table of nine players; each was dealt two cards face-down. Since you don’t have a trey in the hole, then we need consider just the eight opponents.
In all, they were dealt 16 (that’s 8 x 2) cards in the hole out of 50 cards total, deducting your two hole cards. That’s 32 percent of the unseen cards. Then, 32 percent of the four treys equates, on the average, to 1.28 treys among your opponents’ hole cards. In other words, there are less than 3 treys remaining in the deck, available to make your straight.
So you actually have less than three outs – not four as you had used in making your estimate of the card odds. We won’t go through the math, but the result is that your more realistic card odds are about 16-1 against you. With pot odds of 12-1, the card odds are too high to warrant a call.
You could have saved yourself a bet by folding. One might argue that there is still the river card to come if you miss on the turn. Now that’s really gambling and more likely to cost even more if you pursue one of only three – rather than four – treys all the way to the river.
A dollar saved is worth more than a dollar earned. –George “the Engineer” Epstein Do you know why?

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