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A Little MathematicsThere is a widespread error made by most beginning and experienced poker players alike that I want to correct once and for all. It involves figuring the probability of making a particular poker hand when card to come. For instance, if you flop a four-flush in hold’em, what are the chances you will end up with a flush? The above problem is frequently answered incorrectly in the following manner: “I have two spades in my hand and there are two spades on the flop. There are forty-seven cards left and nine of them are spades. My chances of making a flush on the next card are nine out of forty-seven. However, since I have two chances to make the flush-either on the fourth card or the fifth card-my chances double to eighteen out of forty-seven, or 38.3 percent.” Gamblers say in the above case that the player has “nine wins twice.” Unfortunately, most of them think that this is the same as eighteen wins once. It isn’t winning poker strategy. The simplest way to show why you can’t use the above method is by another example. If you pick one card out of a deck, what are the chances that it will be a heart? The answer is, of course, 25 percent. What are the chances of picking a heart if we grab two cards? Your first inclination might be to say 50 percent. But this is wrong. Suppose you pick three cards. Your style of play would say there is a 75 percent chance of picking a heart which is again wrong. The fallacy in this method is completely clear if you picked four cards. By your technique, we would get an answer of 100 percent for the chances of picking a heart. But this is obviously wrong since even picking four cards does not guarantee a heart. What then is the correct method for determining the answer to problems involving questions about a certain number of wins with more than one card to come? The method involves multiplying fractions. The simplest way to answer the problems is to figure the chances that none of the remaining cards dealt will be one of your wins. In the flush draw hold’em problem, it works like this: There are forty-seven unseen cards and nine of them make your flush. Your chance of missing your flush on fourth street is therefore 38/47. Now if you do miss on fourth poker street, there are forty-six cards left and thirty-seven of them miss. You get your chances of missing on both cards by multiplying 38/47 times 37/46. This is 1,406/2, 162. 1,406= (38) (37) This means that you would miss your flush 1,406 out of 2,162 tries on average. You would therefore make your flush 756 out of 2,162 756 = 2,162 – 1,406 or about 35 percent of the time 34.97 = (756) 100 not 38.3 percent as figured by the incorrect method. Let’s try this technique to figure the probability of filling up when you flop two pair not counting the times the odd card on the flop makes trips on the board. You have four wins. Therefore, you have forty-three misses to start. If you miss on fourth street, you now have forty-two misses. Your chances of missing are 1,806/2, 162 1,806 = (43) (42) This leaves a chance of 356 out of 2,162 to make your hand 356 = 2,162 – 1,806 including four-of-a-kind, or about 16.5 percent. 16.47 = (356) 100 Notice that doubling the number of wins to get an incorrect answer is more inaccurate when there are lots of wins. With let’s say, fifteen wins, doubling the wins gives a result of 64 percent rather than the correct answer of 54 percent. When there are more than two cards to come, the shortcut method can be especially bad. Take the case of having a four-flush on fourth street in seven-card stud (assuming no other cards are seen). Of the forty-eight cards left, nine make the flush. Since there are three chances to make it, some people would say the chances are twenty-seven out of forty-eight or about 56.3 percent. The correct method figures the chances of missing three times in a row. It is 54, 834/103, 776. 54,834 = (39) (38) (37) Therefore, you make the flush the remain 48,942 out of 103,776 times, which is 47.2 percent. 48,942 = 103,776 – 54,834 47.16 = (48,942) 100 @Copyright 2005-06 All Rights Reserved www.poker.tj |
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