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The offending post is titled "Poker Probability (Omaha)" and it addresses probability and odds for Omaha. The results they present are clearly wrong and they build upon them. The errors center around the number of unique starting hands.

I have built a database of all possible starting hands. I literally defined every possible unique starting hand and ran a simulation with each one, a minimum of 100,000 times, 10,000 times from each starting position. They claim there are 16,432 possible unique starting hands on page two of a 22 page posting. They build upon this incorrect fact, and while the math is beyond my total comprehension, I do know that 16,432 is incorrect and will prove that the number of unique starting hands is 11,999 beyond a shadow of a doubt.

If you have two independent events and one may be accomplished in m ways and the other in n ways then they may combine is m * n, (read as m times n) ways. For example if we flip two coins there are four possible outcomes, namely H-H, T-T, H-T and T-H. The first coin may land on heads or tails, and the second coin has the same two possible outcomes. Our formula is then m * n, or 2 * 2 = 4. We may apply this to poker in calculating the number of way we may be dealt two aces in hold 'em. We may receive any of the four aces first and any of the three remaining aces second. Following the above rule we would then have 4 * 3 or 12 ways. The chart [not shown] below shows all 12 of the possible ways in which you may be dealt two aces. Looking at the chart one may say, "There is no difference between the starting hand of As-Ad and Ad-As," and you are absolutely correct. The only thing that is different between the two hands is the order. But order does not matter in poker. In fact in most cases a card's suit does not matter unless it relates to a flush or straight flush.

When order does not matter we may still calculate the number of combinations by using the binomial coefficient. The derivation of the mathematical formula for calculating the binomial coefficient is beyond the scope of this article. There is another term we do need to learn and that is factorial. It is denoted with an exclamation point as in n!, which is read "n factorial." The factorial of a number is n*(n-1)*(n-2)... (n-x) until n minus x is equal to one. Therefore 4! is equal to 4*3*2*1 which equal 24, and 2! is equal to 2*1 = 2. To calculate the number of possible combinations to receive 2 aces out of 4, when order does not matter, we simply divide the total number of possible outcomes by the factorial of the number of card to be selected. The symbol "/" will be used for division. Therefore we would have 4*3/2*1 which equals 12/2 = 6.

In English, we have four choices for our first ace and three for our second ace. Remembering the m*n formula we have 4*3 which equals 12, the number we received above that relates to the chart. We now divide this by 2! since we are only taking two cards. 2! equals 2*1 = 2. Dividing 12 by 2 we get the result of 6 combinations in which order does not matter. The chart on the left shows the six possible combinations. There is now only one combination of spades with diamonds because it does not make a difference which card we are deal first.

These are all the tools necessary to verify my calculation of total possible starting hands in Omaha. Remember in Omaha we are dealt 4 cards from a deck of 52 possible cards. Therefore there are 52*51*50*49 or 6,497,400 possible combinations if order matters. Since order does not matter and we are dealt 4 cards we will divide this result by factorial 4! Factorial 4 equals 4*3*2*1 = 24. If we divide 6,497,400 by 24 we get 270,725 possible starting hands.

Next time we will begin calculating the number of unique hands using the techniques above and simple logic.

So what have we learned? You now know how to count the number of possible outcomes taking x number of cards from a deck at a time.

Lastly, here are some more terms from my poker glossary:

Full Bet-A bet that is equal to the proper bet size bet in a limit game.

Full Boat-A hand consisting of any three cards of the same rank plus any two cards of the same rank.

Full House-The same as a full boat. Three-of-a-kind plus a pair.

Full Bet-A bet that meets the minimum required amount set by the limits in a limit game. In pot-limit or no-limit it is the full minimum bet.

Sam Mudaro, BA, MBA, is a practicing tax accountant and financial executive with 35 years of analytical business expertise. He uses simulation software to analyze and develop strategies for Omaha/8 and other forms of poker. Reach Sam at: realguru2003@yahoo.com.