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        A limit poker player can get by with knowing only the approximate odds on a few simple situations. A big-bet poker player needs to know precise information about the odds on a wide range of situations, and how those odds are computed.
       Computing the odds on making a hand with one card to come is quite easy. There are 52 cards in a standard poker deck. Some will be known to you, and most will be unknown. You simply figure the number of unknown cards that make your hand and divide it into the number of unknown cards that leave you a loser. This ratio will be the odds on making the hand.

        Here is an example. On fourth street in a hold’em game the board is K-9-6-4. Your hand is A-3, giving you the nut flush-draw. Suppose you need to hit the flush to win the pot. What are your chances?
        In this problem six cards are known to you, leaving 46 unknown cards (52 minus 6 equals 46). Since there are 13 hearts in a deck of cards, and you have a four-flush, there are 9 hearts left for you to catch for the flush. That means out of the 46 unknown cards, 9 make the flush, and 37 do not (46 minus 9 equals 37). Therefore, the odds are 37 to9 against making the flush. Another way to say this is you have 9 chances out of 46 to make the flush. In other words, the chances are just over 4-1 against you, or slightly worse than one out of five.
        Suppose the game in the above situation was pot-limit hold’em, and on fourth street the opponent bet the size of the pot. Could you call on this hand?  The pot odds offered you are a mere 2-1, because the amount in the pot is double the amount it costs you to call. For example, if there is $1000 in the pot and the opponent bets a grand, you are getting 2-1 pot odds on the grand it costs you to call. Since it is over 4-1 against you, this would be a bad call. In a sense, you would be making about a $500 mistake.

        If there is money left to be bet after making your draw, it is possible that the money you might make after hitting the flush could justify a call, even though you were not getting the right price. We use the term “implied odds” when our calculations have included additional money we think can be won in further betting. Let us see how the “implied odds” concept can be used.
        Suppose in our flush-draw case the opponent had bet only $500, half the size of the pot. Should we call now? The odds have changed. It will cost us $500 to win $1500. We are getting 3-1 money odds and are a slightly more than a 4-1 underdog, so the price is still wrong for a call, if you look at the pot odds. But if we think there is a reasonable chance of making some money if we make the flush, it could be right to call. Even after we know how to compute the math, we still have to use other poker skills to see whether the implied odds justify a call. In this particular situation, because it is a community card type of game, the opponent will see three hearts on the board if we make out hand. He will likely be nervous at the looks of that last card. At limit play the opponent often grits his teeth and calls your bet, because the terrific pot odds he is getting means he cannot afford to fold a possible winner. At pot-limit or no-limit play, it is a lot harder to get the opponent to pay us off when we make a hand, especially when it is something as obvious as making a flush. Still, you must think about how the betting has gone to this point, who the opponent is, whether he is stuck in the game, and any other factor that could affect the implied odds. Most decisions at big-bet poker involving the odds will require, in addition to a proper knowledge of the actual odds, good judgment in assessing the implied odds. Some aspects of figuring implied odds were discussed in out chapter titled “Key Pot-limit Concepts.”

         The implied odds do not always make a call more attractive. Sometimes the real odds on a drawing hand will be worse than they first appear. For example, in our previously mentioned layout, where out holding is A-3 and the board is K-9-6-4, it is possible that we could hit the flush and still lose. Note that two of our possible flush-cards, the 6 and the 4, also pair the board. If the opponent has a set or the fitting two pair, he will fill up when we hit out flush with either of those board-pairing flush-cards. This means that we may have only seven outs instead of nine. Also, we get to lose mucho bucks if a card hits both us and the opponent.
          Naturally, if we make our hand with one card still to come, the math may change drastically. There are now two betting round instead of one on which we may make money. There is also the possibility the opponent may redraw on the last card and beat us. This is especially important when drawing at a straight, when either a flush or a full house will beat us on the end. On the other hand, straights have a greater concealment factor, which normally enables you to get them paid off more easily than a flush or a full house.

          We have used the term “implied odds” to talk about additional money beyond the present pot size that might be won or lost if we make out hand. There is another factor that must be taken into consideration when deciding whether to call a bet. By calling, we are still contending for the pot. As we know, besides making the best hand, we may also win the pot us the opportunity to launch a bluff. Retention of “bluffing right” is of enormous importance at big-bet poker, and must be included when calculating whether to call the opponent. 
          Here is an example from no-limit hold’em of a situation where “bluffing rights” are of paramount importance. You pick up Q-K under the gun and call the blind. The small blind calls and you play a threehanded pot vs. the small and big blinds. The flop comes A-J-6 of three different suits, giving you a gutshot ten draw for a straight. The first player bets the size of the pot, the second player folds, and it is up to you. Let’s say there is $30 in the pot and the bet is also $30. Should you call?  


The math of the situation is there are 47 unknown cards, of which 4 give you a winner and 43 are quite likely to leave you in second place. The odds are nearly 11-1 against you. Yet the fact of the of the matter is a call in this type of situation is quite a reasonable play and not uncommon at big-bet poker. To see why, let’s look at the deal through your opponent’s eyes. He has probably got an ace with a weak kicker for his bet. With a good kicker, he may well have raised preflop. He could have flopped two pair, but it is more likely he has only one pair. For your call under the gun preflop and a subsequent call of his bet, he will thinks you probably have an ace with a decent-size kicker. His most likely course of action is to check on fourth street if he still has only one pair, and release his hand if you bet. This means your game plan is to call on the flop and take the pot away from him if he checks to you on fourth street. It is a sound plan against many players, even though the odds on actually making the hand are hugely unfavorable. “Bluffing rights” must be taken into consideration when weighing whether to call, if there is the possibility of more betting on the deal.

        Calculating the odds with two or more cards to come is far more difficult than doing so with only one card to come. For example, after the flop in a hold’em game there are ordinarily 47 unknown cards. (The 52-card deck less the two cards in our hand and the three board-cards equals 47.) After fourth street is dealt, there will be 46 unknown cards that could come on the end. Thus the total number of card combinations for the last two cards is 47 times 46, which is 2,162. The only accurate way to calculate the odds on something such as making a flush is to count up all the card combinations that make the hand and compare it with that total-combination number of 2,162.

       Here is how we find the true poker odds on making the flush. We make it on fourth street with 9 cards. We multiply 9 times 46 (the number of unknown cards that could come on fifth street) to give us the number 414. There are 38 card that do not complete the flush on fourth poker street (the 47 unknown cards minus the 9 cards that complete the flush). To get the number of combinations that make the hand on fifth street, we multiply 38 times 9 (the number of cards that complete the flush on fifth street) which gives us 342. That number is all the card combinations that make the hand on fifth street we did not already have the hand on fourth street. It is important to note why we multiplied 38 times 9 rather than 47 times 9. It only helps us to make the hand the on fifth street when we have not yet made the flush. There is no bonus for making the hand on fourth street and again on fifth street. (Indeed, if we are not drawing at the nut flush, and are playing Texas hold’em rather than Omaha, it could be quite detrimental to hit a heart on both fourth and fifth street.) To get the total number of card combinations that make our flush, we add 414 and 342 to get 756. This means that the odds on making our flush are 756 out of 2,162. Another way to say it is the odds are 1406 to 756 against us (the number 1406 is 2,162 minus 756).

       The method of computing the odds of making a draw with two cards to come is quite messy. No big-bet player does it at the table. Rather, we are familiar with the odds on certain situations that are quite commonplace at the form of poker we favor. All good hold’em players know they are slightly less than a 2-1 underdog to hit a flush with two cards to come.
        We often use shortcuts to give us a ballpark figure that is close enough to the true odds for most poker purposes. For example, if a t hold’em I have pocket aces and my opponent has a set, here is how I would get an approximate idea of my chances of helping with two cards to come. There are 45 unknown cards that could come on fourth street. (My two cards, the opponents two cards, and the three boardcards are known; 52 minus 7 gives us 45.) Of these 45 cards, only 2 help me, and 43 do not. Thus the odds on making the hand on fourth street are 43 to 2 against. That is 21.5 to 1. Since there is yet another card to come, we can take half of 21.5 which is 10.75. The approximate number I get with this method is 10.75 to1 against me making three aces.
        Here is how the true figure is computed. The number of ways to make the hand on either card is [(2 times 44) plus (43 times2)], which is 174. The total number of possible card-combinations is 45 times 44, which is 1980. My change of making the hand is thus 174 out of 1980, so 174 card-combinations make the hand and 1806 do not. The odds on my making the hand are 1806 to 174 against me, which reduces to 10.38 to 1. 
        As you see, in this case we got a reasonable ballpark figure for figuring the odds on making the hand with two cards to come by guessing that they were about  twice as good as odds on making the hand with one card to come. In this case the crude figure was 10.75 to1 and the real number was 10.38 to 1. The true odds with two cards to come are microscopically better here for the person drawing than the figure derived by the crude method, but the numbers are so close that the decision of whether to call is not going to be affected by the discrepancy between the methods used.

        So the simple but crude method to find the chance of hitting with two cards to come is to take the chance of hitting the hand with one card to come and say your chance is about twice as good with two cards to come. Expressed in concrete terms, if you are a 20/1 dog on the next card you will be only about a 10/1 dog with two cards to come. If you are a 6/1 dog on the next card you will be only about a 3/1 dog with two cards to come.
         The above method worked reasonably accurately because we were figuring the price on a longshot. This method is notoriously inaccurate when the draw is not a longshot. For example, it is obvious that a draw that is even money with one card to come will not be a cinch if there are two cards to come. In other words, a 5 percent draw with one card to come increases to almost 10 percent with two cards to come, but a 50 percent draw with one card to come does not go to anywhere near 100 percent when there are two cards to come. The right way to think about an even-money draw is about half the time you hit it on the end you already had made it on fourth street. Thus the chances rise from 50 percent with one card to come to about 75 percent with two cards to come.

          The mathematical aspect of big-bet poker is quite complex. You must acquire a feel for figuring the odds in the more common type of drawing situations. But equally important is having a feel for the implied odds of a situation, and knowing how much “bluffing rights” are worth in a particular spot. Only by taking into account all these factors can you give yourself the best chance for a successful decision.


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