During the 2008 WSOP Main Event, quad aces lost to a royal flush with both hole cards playing. (You can view the two-minute video at http://www.youtube.com/watch?v=XunAlp2azhA)

In post-production ESPN’s math consultant rummaged around in a place renowned for its lack of sunshine and pulled out a number for the odds. In the show’s airing, ESPN’s Lon McEachern announced that number to the world, 2.7 billion-to-1. Wrong. Nope. Nyet. Non. No way. Not even close—that number misses by two orders of magnitude! Let’s calculate the right odds.

We break the problem into three parts: how many boards allow both quad aces and a royal when both players’ hole cards must play; all four key cards must lie in the top part of the deck; and, the four key cards must lie in two players’ hands arranged correctly.

Four ways to put an ace on the table, times six ways to put two of the four royal cards of that suit on the table, times three ways to put another ace on the table, times the number of fifth cards to place on the table, equals 4*6*3*44, the number of boards where four aces could lose to a royal flush. We divide that number, 3,168, by the total number of five-card boards possible, 2,598,960, to find the probability of a qualifying board, 0.00122.
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The dealer had to deal the other two aces and the other two royals among the 2*P cards dealt to the players. The expression, C(43, 2*P-4)/C(47, 2*P), obtains the probability that all four key cards lie among the number of players, P.

Those four cards must lie correctly in two players’ hands. We obtain that probability from the expression, (2*P-5)!!/(2*P-1)!!, where again P equals the number of players dealt into the hand. The doubled quotation marks denote the product of every odd number from the starting number down. For a nine-handed game, that probability equals (13*11*…*3*1)/(17*15*13*…*3*1), which equals 1/255.

Multiplying the three probabilities yields the probability of quad aces losing to a royal flush at a hold ’em table. We put all that into a spreadsheet and computed the probability for each number of players at the table from 2 to 11. (E-mail me for a copy of the large spreadsheet; it’s too large to fit in this column.)

At a two-player table the odds against four aces losing to a royal equal 439 million-to-1 against. At an eleven-player table the odds equal 8 million-to-1. As nearly as we can tell from YouTube’s video, this event occurred at a nine-player table: the odds against that are 12.2 million-to-1.

At odds against of 12.2 million-to-1, a nine-player table has a 50-50 chance of having quad aces lose to a royal within the next 8.5 million hands. At 100,000 hands per year for a typically avid hold ’em player at a brick and mortar poker room, you have a 50-50 chance of seeing such a deal within the next 85 years.

On the internet, because of its speed of play, you might see such a deal in a lifetime. At 100,000 hands per year, you personally have a 50-50 chance of holding either hand within the next 380 years. The current popularity of poker suggests that such an event occurs weekly somewhere in the real world or in cyberspace.

Still, at odds against of 12 million-to-1 ESPN lucked out in catching the event live. Why announce a much larger number for the odds? Either ESPN’s math consultant made serious logic errors, or the ESPN’s producers wanted a spectacular, BIG number for the show. We suspect both.

Mr. Burke is the author of Flop: The Art of Winning at Low-Limit Hold ’Em, on sale at amazon.com & kokopellipress.com. E-mail your Hold ’Em questions to richardburke@comcast.net