Poker News

Richard Burke

After signing up for a $4-8 Hold’Em game, LindaMae strode over to my chair, sat, and asked, “What is that C thingy I see in your articles?” Cleverly, I responded, “C thingy?” Yes, she said that it was in all those formulas that clutter up my columns.

Finally I understood. I toldher that the C thingy was shorthand for the number of ways that items could be taken from a set. C(n,m) means n!/(n-m)!/ n!, where n and m are integers, and m <= n. When I showed her that, she asked why all the exclamation points. Those were not exclamation points, I told her, they meant that the numbers were factorial. For example, C(5,2) really means 5*4*3*2*1/3*2*1/2*1: that there are 10 ways to select two items from a set of five. That C thingy is fundamental to combinatorial mathematics, I told her.

How many possible starting hands are there in Hold’Em, I asked her. LindaMae informed me that she had majored in Psychology and had deliberately stayed away from mathematics. There are 52 cards in a standard deck, and you need two in your hand. The number of possible hands, counting suits, is C(52,2), or 52!/50!/2!, which simplifies to 1326. That’s the number of possible starting hands in Hold’Em.

How often will your starting hand be paired, I asked. Hearing no answer, I said the dealer must deal you two cards from any of the thirteen ranks of four cards each. The number of pocket pairs possible is 13*C(4,2), which simplifies to 78. Your hand will be paired this fraction of the time, 78/1326, which simplifies to 1/17. One time in 17 you’ll pick up a pocket pair. LindaMae said she knew that; everyoneknew that. Possibly, I said, and now you know why. How often will your starting hand be suited, I asked her. There are four suits of thirteen cards, and you need two from any of the thirteen, so there are 4*C(13,2) possible suited starting hands, which simplifies to 312. The chance that your starting hand will be suited is 312/1326, which simplifies to 4/17, or .2353. Your starting hand will be suited a little less than 25% of the time. She nodded.

How often will you be dealt two cards to a Royal Flush, I asked. Not waiting for an answer, I said there are only four Royals possible, so the number is given by 4*C(5,2), which equals 40. The chance of being dealt two-to-a-Royal is 40/1326, .0302. About 3% of the time, you’ll pick up two-to-a-Royal. She nodded.

You need that C thingy to solve many other gambling problems, not just cards. What is the chance of hitting two numbers in a Keno game, I asked. There are twenty ping-pong balls selected randomly from a set of eighty. You must hit both from the two you choose, C(2,2). The other eighteen must come from the 78 other ping-pong balls: that’s C(2,2)*C(78,18), which is a very large number, 2.13*1017. The catch is the total possible ways to select twenty from eighty, C(80,20), is even larger, 3.54*1018. Your chance of hitting both is the quotient, which simplifies to 19/316, or .0601. You have about a 6% chance of hitting two out of two Keno numbers, roughly one time in 16.6 trials.

Do you see now how important that C thingy is, I asked. There was no response: she had nodded off.

Filed under: Poker News