# Theoretical Probability of Winning Craps

The game of craps is unique in a couple of ways. For one thing, the game offers some of the best bets in the casino. For another, it also offers some of the worst bets at the same time. Most casino games either have a high house edge or a low house edge; craps has both.

Another intriguing aspect of craps is that it’s the only casino game where you can bet on something NOT happening. In all other casino games, the only event that you can bet on is the one that happens. You can’t bet that red won’t come up in roulette; you can only bet on black or on one of the green zeroes.

Both of these make for interesting uses of probability in analyzing the game.

## Possible Totals and Their Probabilities

In craps, there are only 12 possible totals, but the probabilities of the various totals vary significantly. That’s because, as a probability problem, each total (except for 2 and 12) can be achieved in multiple ways.

Here’s a list of each of the totals along with the ways of achieving them and the probability of getting each of them:

2 is only possible if you get a 1 on both dice. The probability of that happening is 1/36. This is calculated by multiplying the odds of getting a 1 on the first die with the probability of getting a 1 on the second die. As a math problem, it looks like this: 1/6 X 1/6 = 36.

3 is possible in 2 different ways. The first is to get a 1 on the first die and a 2 on the second die, and the second is to get a 2 on the first die and a 1 on the second die. This makes a total of 3 twice as likely as a total of 2, which is a probability of 2/36.

4 is possible in 3 different ways. You can get a 1 on the first die and 3 on the second. You could also get a 3 on the first die and a 1on on the second. Or you could get a 2 on both dice. That’s 3 times as likely as getting a 2, which means the probability is 3/36.

5 is possible in 4 different ways. You can get a 1 and a 4, a 4 and a 1, a 2 and a 3, or a 3 and a 2. That’s 4 times as likely as getting a 2, which means the probability is 4/36.

6 is possible in 5 different ways. You can get a 1 and a 5, a 5 and a 1, a 2 and a 4, a 4 and a 2, or a 3 and 3. That’s 5 times as likely as getting a 2, which means the probability is 5/36.

7 is the most likely outcome when rolling two dice. There are 6 ways to get a total of 7: a 1 and a 6, a 6 and a 1, a 2 and 5, a 5 and a 2, a 4 and a 3, and a 3 and a 4. This is 6 times as likely as rolling a 2, which means the probability is 6/36.

The rest of the totals correspond accordingly. A total of 8 has the same probability as a total of 6. A total of 9 has the same probability as a total of 5. A total of 10 has the same probability as a total of 4. A total of 11 has the same probability as a total of 3, and a total of 12 is exactly as likely as a total of 2.

## Expressing these Probabilities

Expressing a probability as a fraction is just one way to express it. Another way is to express a probability as a decimal. You could also convert that to a percentage, which is very common and quite intuitive. Finally, you can express probabilities in odds format.

Understanding that something will happen once every 36 times is intuitive enough when looked at as a fraction. But 2.78% is also an intuitive way of looking at it. Poker players probably prefer to look at that number as odds, which are 35 to 1. (There are 35 ways to NOT roll a 2, and only 1 way to roll the 2.)

## Understanding the House Edge

Probability matters because of the house edge. Casino games are rigged, as it turns out, but not in the way you think. The reason casinos are profitable is because they never pay bets out at their true odds of happening.

For example, if you place a bet that a 2 is going to come up on the next roll of the dice, the odds of winning are 35 to 1. But this bet pays off at 30 to 1.

So suppose you bet \$1 on that total 36 times in a row, and you saw mathematically perfect results. (This won’t happen in the short run, but the more rolls of the dice that are seen, the closer the results will mirror the mathematical expectation.)

You’ve bet a total of \$36. You won one of those bets, and you got \$30 out of the deal. The difference of \$6 is your net loss. That difference represents the house edge, and it’s usually expressed as a percentage. The house edge on this particular bet is 13.89%, which means that you can expect to lose \$13.89 for every \$100 you wager.

This number might be meaningless, but you can use it to calculate how much entertainment you see for your dollar. The reality is that if you play any game with a house edge long enough, you’ll eventually lose all your money to the house. The trick is to maximize the amount of entertainment you get out of the deal.

## Calculating Expected Losses per Hour

Suppose you’re playing craps, and you’re making the bet on the 2 over and over. The house edge is 13.89%. Assume that you’re able to make 40 bets per hour at \$2.50 per bet. That’s \$100 per hour you’re putting into action.

At that rate, you’ll lose \$13.89/hour over the long run.

Compare that with another game, roulette. Let’s say that you’re playing on a standard American roulette wheel, where the house edge is 5.26%.  You can assume 40 bets per hour here, too, and if you assume the same amount per bet, you’re looking at losing \$5.26 per hour over the long run.

Roulette seems like the better game, right? But that’s just because we picked one of the worst bets at the craps table.

The best bet at the craps table is the don’t pass bet, which has a house edge of only 1.36%. Now you’re looking at an expected loss of \$1.36 per hour. Heck, you can drink a couple of free cocktails an hour and wind up having a lot of fun for very little money at that rate.

Understanding probability and the house edge is the first step to becoming an intelligent gambler.