Hack 36. Know When to Hold 'Em
In Texas Hold 'Em, the "rule of four" uses simple counting to estimate the chance that you are going to win all those chips.
Texas Hold 'Em No Limit Poker is everywhere. As I write this, I could point my satellite dish to ESPN, ESPN2, ESPN Classics, FOX Sports, Bravo, or E! and see professional poker players, lucky amateurs, major celebrities, minor celebrities, and even (Lord help us, on the Speed channel) NASCAR drivers playing this simple game.
You probably play yourself, or at least watch. The most popular version of the game is simple. All players start with the same amount of chips. When their chips are gone, so are they. Every round, players get two cards each that only they (and the patented tiny little poker table cameras) see. Then, three community cards are dealt face up. This is the flop. Another community card is then dealt face up. That's the turn. Finally, one more community card, the river, is dealt face up. Betting occurs at each stage. Players use any five of the seven cards (five community cards, plus the two they have in their hands) to make the best five-card poker hand they can. The best hand wins.
Because some cards are face up, players have information. They also know which cards they have in their own hands, which is more information. They also know the distribution of all cards in a standard 52-card deck. All this information about a known distribution of values [Hack #1] makes Texas Hold 'Em a good opportunity to stat hack all over the place [Hacks #36 and #38].
One particularly crucial decision point is the round of betting right after the flop. There are two more cards to come that might or might not improve your hand. If you don't already have the nuts (the best possible hand), it would be nice to know what the chances are that you will improve your hand on the next two cards. The rule of four allows you to easily and fairly accurately estimate those chances.
How It Works
The rule of four works like this. Count the number of cards (without moving your lips) that could come off of the deck that would help your hand. Multiply that number by four. That product will be the percent chance that you will get one or more of those cards.
You have a Jack of Diamonds and a Three of Diamonds. The flop comes King of Clubs, Six of Diamonds, and Ten of Diamonds. You have four cards toward a flush, and there are nine cards that would give you that flush. Other cards could help you, certainly (a Jack would give you a pair of Jacks, for example), but not in a way that would make you feel good about your chances of winning.
So, nine cards will help you. The rule of four estimates that you have a 36 percent chance of making that flush on either the turn or the river (9x4 = 36). So, you have about a one out of three chance. If you can keep playing without risking too much of your stack, you should probably stay in the hand.
You have an Ace of Diamonds and a Two of Clubs. The flop brings the King of Hearts, the Four of Spades, and the Seven of Diamonds. You could count six cards that would help you: any of the three Aces or any of the three Twos. A pair of twos would likely just mean trouble if you bet until the end, so let's say there are three cards, the Aces, that you hope to see. You have just a 12 percent chance (3x4 = 12). Fold 'em.
Why It Works
The math involved here rounds off some important values to make the rule simple. The thinking goes like this. There are about 50 cards left in the deck. (More precisely, there are 47 cards that you haven't seen). When drawing any one card, your chances of drawing the card you want [Hack #3] is that number divided by 50.
Whatever that probability is, the thinking goes, it should be doubled because you are drawing twice.
For the first example, the rule of four estimates a 36 percent chance of making that flush. The actual probability is 35 percent. In fact, the estimated and actual percent chance using the rule of four tends to differ by a couple percentage points in either direction.
Other Places It Works
Notice that this method also works with just one card left to go, but in that case, the rule would be called the rule of two. Add up the cards you want and multiply by two to get a fairly accurate estimate of your chances with just the river remaining. This estimate will be off by about two percentage points in most cases, so statistically savvy poker players call this the rule of two plus two.
Where It Doesn't Work
The rule of four will be off by quite a bit as the number of cards that will help you increases. It is fairly accurate with 12 outs (cards that will help), where the actual chance of drawing one of those cards is 45 percent and the rule of four estimate is 48 percent, but the rule starts to overestimate quite a bit when you have more than 12 cards that can help your hand.
To prove this to yourself without doing the calculations, imagine that there are 25 cards (out of 47) that could help you. That's a great spot to be in (and right now I can't think of a scenario that would produce so many outs), but the rule of four says that you have a 100 percent chance of drawing one of those cards. You know that's not right. After all, there are 22 cards you could draw that don't help you at all. The real chance is 79 percent. Of course, making a miscalculation in this situation is unlikely to hurt you. Under either estimate, you'd be nuts to fold.