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A Little mathematics

There is a common error made by most beginning and experienced players which has to be corrected. It involves figuring the probability of making a particular online Omaha poker hand when one more card will make the hand but there is more than one card to come. For example if you flop four-flush in hold'em, what are the best possible chances you will end up with a flush?

The above problem is often 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 spades. My chances of making a flush on the next card are nine out of forty-seven. However, as 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."

Most of the payers say in the above case that the players has "nine wins twice." But many of them think that this is the same as eighteen wins once. It isn't.

The simple 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 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. For instance you pick three cards. Your inclination would say there is a 75 percent chance of picking a heart which is again wrong.

The fallacy is clear if you picked four poker cards. By your technique, we would get an answer of 100 percent for the chances of picking a heart. But this is wrong since even picking four cards does not warrant a heart.

What then is the correct method to determine the answer to problems involving questions about a given number of wins with more than one card to come? The method involves multiplying the fractions. The simple way to answer the problem is to identify the chances that none of the remaining cards dealt will be one of your wins.

In the flush hold'em problem, it works as follows: There are forty-seven unseen cards and nine of them make your flush. Your chance of missing your flush on fourth street is this 38/47. Now if you miss on the fourth 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 1406/2162.

1406/2162 = 38/47 * 37/46

This means that you would miss your flush 1406 out of 2162 tries on average. You would therefore make your flush 756 out of 2162.

2162 - 1406 = 756

or about 35 percent of the time

34.97 = 756/2162 * 100

not 38.3 percent as figure by the incorrect method.

Let's try this technique poker technical 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. Thus, you have forty-three misses to start. If you miss in fourth street, you now have forty-two misses. Your chances of missing are 1806/2162.

1806/2162 = 43/47 * 42/46

This gives a chance of 356 out of 2162 to make your hand.

2162 - 1806 = 356

including four of a kind, or about 16.5 percent.

16.47 = 356/2162 * 100

Notice that doubling the number of wins to get an incorrect answer is more inaccurate when there are more wins. With let's say, fifteen wins, doubling the wins gives the answer 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 truly bad. Take the case of having a four-flush on fourth poker street in seven card stud (assuming no other cards are seen). Of the forty eight cards left, nine make the flush. As 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 54834/103776.

54834/103776 = 39/48 * 38/47 * 37/46

Thus you make the flushes the remaining 48942 out of 103776 times which is 47.2 percent.

48942 = 103776 - 54834

47.16 = 48942/103776 * 100

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