Nothing's a Lock

"Jack of diamonds, jack of diamonds, jack of diamonds," I cry …

When reading the Internet poker forums, you see a lot of complaining about the bad beats that happen on one poker site or another. "This kind of thing is impossible; it doesn't happen this way in real life." "I had XX and my opponent had YY, and there's no way he could catch up to me, but he did. The cards are obviously rigged." And you see the same thing when you actually play online; the same comments come up in chat: "That river card could have come here only on this site." I note with more than a little amusement that I've seen that comment on every poker site I've ever played, thus proving that these bizarre river cards are indeed limited to – er – all online poker sites.

There are several reasons why people make these comments, but I'd like to discuss a few of them right here:

First, you see many more hands online than you do in a brick-and-mortar casino or cardroom. Consider this: Suppose you're playing in a cardroom with a pretty snappy dealer and players are acting promptly. Nobody is asking for deck changes too often, and things are running smoothly. You're seeing about 30-35 hands per hour. Now, contrast that to an online environment. Suppose you're playing two games at once, at a very reasonable 80 hands per hour each. So, you're seeing 160 hands per hour – five times as many hands as you do in your local cardroom. And you will see five times as many wacky events, five times as many bad beats, and five times as many monster hands. ("Quads should never happen that frequently!") As live-game poker players, we're conditioned to expect to see certain things with a specific frequency. For instance, if you play a lot of cardroom hold'em, you might make a straight flush with both of your cards once a year – and you remember them. Now, you're doing it every two months and they all blur together.

Second, people seem to get a distorted idea of how frequently the best hand is supposed to win. For instance, consider the classic "dominated hand" scenario – A-K vs. A-Q. Whenever the A-Q wins, people go nuts; it wasn't fair, the site is rigged, and so on. Try the following experiment: Get a pair of dice. If necessary, borrow them from the house Monopoly game. Start rolling. Every time you see 2, 3, 4, 11, or 12, A-Q beats A-K. That's right, sports fans, A-K is about a 3-1 favorite over A-Q all in preflop (if they have no matching suits). So, A-Q gets there 25 percent of the time. With your dice, you've got 36 (6 × 6) possible combinations. One combination makes 2, two combinations make 3, three make 4, two make 11, and one makes 12 – a total of nine of the 36 possible combinations. If it will make you feel better, every time you roll 2, 3, 4, 11, or 12, yell, "These dice are rigged!"

My friend Bill Chen has a knack for explaining mathematical concepts so that we laypersons, who think math is hard, can understand them. He points out that A-Q beats A-K with approximately the same frequency that a big league ballplayer gets a base hit (if he doesn't walk). Tell the pitchers facing a hitter that it's a statistical impossibility when he gets a hit.

And it's not hard to come up with other examples of "impossible" outcomes that really do happen. For instance, suppose you push all in with the 7 6 and run smack into two black aces. It's time to pick up your bottle of water and go home, right? Well, probably, but it's not as hopeless as you might think. You have about a 23 percent chance of winning the hand. Still got those dice out? Try this: There are four combinations each of the fives and nines, and 8?36 is almost exactly the chance that your 7-6 suited has against those ugly aces. Fire up the dice again. Every time you roll a 5 or 9, the aces get cracked. It doesn't happen much, but it happens, for sure.

You might find it worthwhile to try other experiments such as those above. It will help you get a perspective on how often these "impossible" things actually happen. First, you'll need those dice. And, you'll need something to tell you what the odds of interest are. One good poker odds calculator is at www.cardplayer.com.

Once you find the odds of the matchup you want, you need to know how to convert that into dice probabilities. As I mentioned, there are 36 possible combinations of a pair of dice. It's easy to count them, but for convenience, there are:

• one each for 2, 12

• two each for 3, 11

• three each for 4, 10

• four each for 5, 9

• five each for 6, 8

• six for 7

So, now just multiply 36 by the chance that your underdog hand has. For instance, A-10 offsuit is just a 3-2 favorite over K-Q offsuit (with four suits among them). The K-Q has a 40 percent chance. That's 14.4 "outs" out of 36. There are five combinations each of the 6 and 8, and four combinations of the 5. If you want to make it a bit more accurate, give the K-Q any 9 made of 5+4.

Just don't blame me if your husband walks into the kitchen as you throw the dice, they come up 3+5, and you say, "My K-Q just sucked out on A-10!"

Here's one final thing: The order that the cards come out doesn't matter. It's perhaps more painful for the pocket jacks when the king that hits the A-K comes on the river rather than on the flop. But remember that the river card was decided when the money went into the pot. I knew a dealer at a club nearby who used to (with the players' permission) deal the entire board facedown before exposing any cards when there was no more action. He'd burn and deal the flop facedown. Then, he'd burn and deal the turn facedown, and do the same with the river card. Then, he'd grab all five cards and turn 'em faceup. There was no agonizing over the river card, as it came out right with the flop. When you think about it that way, you realize that he didn't change a single thing about the outcome – just the way we experienced it. Maybe you prefer the suspense and excitement of the successive cards. That's fine – but when the river beats you rather than the flop, don't forget that it was all rolled into those same probabilities.

So, you don't need to rig poker decks to get "bizarre" results. A reasonably shuffled deck of cards will do that for you.

"If I don't get the jack of diamonds, I surely will die."