Craps

Craps is a dice rolling game. There are dozens of bets but we're only interested in two: betting the pass line and taking the odds. Nearly all of the other bets are for suckers. You can read descriptions and analysis of all craps bets on Wikipedia.

There are two six-sided dice. You may only bet on the pass line before the first roll. If the shooter (dice roller) rolls 7 or 11, you win. If he rolls 2, 3 or 12 then you lose. If he rolls any other number (4, 5, 6, 8, 9, 10) then that number is established as the point and the game continues. The shooter rolls repeatedly until either he rolls the point again, winning, or 7, losing.

Expected Value

The expected value of gambling is the product of the payout¹ and the probability of winning that payout. If you win the pass line bet in craps then you double your money. I.e., the payout is 2:1. The chance of winning the pass line bet is exactly $\frac{244}{495}$ or about 49.3%. The expected value is twice that or about 98.6%. You can imagine it this way: if you bring $1,000 to Las Vegas and make 10, $100 pass line bets, you would expect to walk away with around $986.

+
	2	3	4	5	6	7
	3	4	5	6	7	8
	4	5	6	7	8	9
	5	6	7	8	9	10
	6	7	8	9	10	11
	7	8	9	10	11	12

Where does that $\frac{244}{495}$ number come from? I'm so glad you asked! The way to calculate probability is to count up all the possibilities and then count how many of those outcomes are winners. For two six-sided dice, there are $6^{2} = 36$ possible rolls. I summarized them in the table to the left so you can count with me. There are 8 ways to win instantly (7 or 11).

Counting up the points is lengthy but not hard. We can cut the work in half by noticing the diagonal symmetry in the table. There are an equal number of 4s as 10s, 5s as 9s and 6s as 8s. So, we can just count the ways to win when the point is a 4, 5 or 6 and then multiply by 2.

There is a $\frac{3}{36}$ chance of 4 being the point. The goal is to roll another 4 before rolling a 7. There are 3 ways to roll a 4 and 6 ways to roll a 7. Thus, there is a $\frac{3}{9}$ chance of winning once 4 is established as the point. There is a $\frac{4}{10}$ and $\frac{5}{11}$ chance of winning when the point is 5 or 6, respectively. Putting it all together:

\frac{8}{36} + 2 (\frac{3}{36} \cdot \frac{3}{9} + \frac{4}{36} \cdot \frac{4}{10} + \frac{5}{36} \cdot \frac{5}{11}) = \frac{8}{36} + \frac{134}{495} = \frac{244}{495}

At 98.6% expected payout, the pass line is one of the best bets in Las Vegas. But wait, it gets even better! If a point is established, you get the option to take the odds. Physically, you place a little stack of chips behind your bet on the pass line. It's not marked on the felt, almost like the casino is hiding something…

Your odds bet wins if your pass line bet wins. It pays 3:1 when the point is 4 or 10, 5:2 when the point is 5 or 9 and 11:5 when the point is 6 or 8. The expected value of taking the odds is respectively $\frac{3}{9} \cdot \frac{3}{1} = \frac{4}{10} \cdot \frac{5}{2} = \frac{5}{11} \cdot \frac{11}{5} = 1$ . Yes, the expected payout is 100%. Let that sink in. Taking the odds is the only fair bet in Las Vegas (but you must make the unfair pass line bet first).

Most casinos have so-called 3-4-5 odds. That means you can bet up to 3 times your pass line bet when the point is 4 or 10, 4 times when the point is 5 or 9 and 5 times when it's 6 or 8. There's a practical reason for this: no matter what the point is, the boxman only has to remember to give you 6 chips when you win. If you always take the maximum odds, and of course you should, then the expected value of craps is a mighty $\frac{1863}{1870}$ or about 99.626%.

Variance

Gambling is fun because you can win in the short run. That's known as variance. A game without variance would be no fun, no matter how tight the spread. Imagine playing a game with 99.9% payback but no variance. You would put $10 on the table then the boxman would immediately give you $9.99 back. It may only be one penny but that's no bargain at all!

The greater the variance, the more time you have "before the odds catch up to you." That could be good, as in winning lots of money, or it could be bad, as in burning through your bankroll² quickly. True gamblers play games with high variance. Social gamblers tend to play games with low variance because they're after a night of entertainment, not winnings per se. A high variance game might burn through their bankrolls too quickly and send them home early.

The roulette wheel is the best place to see variance in action because it offers high and low variance at equal expected value.

The expected value of betting on a color or a number is the same: $2 \cdot \frac{18}{38} = 36 \cdot \frac{1}{38} = \frac{18}{19}$ or about 94.7%. If you bet on a number and win (2.63% probability) then you will have multiplied your bankroll 36 times. On the other hand, attempting to reach the same multiple by placing color bets is approximately impossible. The problem with betting it all on a number is that you'll be headed home after a single spin, before free drinks arrive!

The variance in craps occupies a happy middle ground. It's high enough that you can plausibly win a lot of money yet low enough that you won't burn through your bankroll too quickly.

You must establish a couple of numbers before you head for the tables:

your bankroll: the maximum amount of money you're willing to lose
your goal: you walk away when you reach your goal

People often overlook establishing a goal. People tend to keep doing what feels good and winning feels good. The problem is that if you play for long enough, you will lose. The fewer bets that you place, the more you expect to walk away with. Good ideas include not gambling at all or betting your entire bankroll at once. Some people wait until the last day of their vacation, place a single bet then catch a cab to the airport.

Not gambling, or betting once may fail to maximize excitement. You only get to experience gambling for a few seconds. That's not enough time drink booze and surf the waves of probability. I wanted to find a reasonable amount to bet per game of craps. The amount should be high enough that I have a plausible chance to reach my goal but be low enough so I get to roll the dice a few times.

Here's the program that I wrote to simulate playing craps. Give it your bankroll, goal and pass line bet. The program will assume you take maximum odds on a 3-4-5 table. You can either get a detailed readout for a single simulation or some averages for 10,000 simulations. I found that trying to double your bankroll by betting 5-10% of it on the pass line each game is a good compromise between fun and winning.

Bankroll
Goal
Pass Line Wager

Notes

Most tables express the payout in terms of how much the casino pays you, in addition to your original bet. I am instead expressing payouts in the mathematical style where your original bet is included in the figure. For example, my 3:1 payout would be called a 2:1 payout in a casino. Using the mathematical style makes the calculations easier to follow.
Your bankroll is the maximum amount of money that you are willing to lose. It is irrational to buy in for more than your bankroll because there is no point in having chips that you never intend to bet. Similarly, it's irrational to walk away from a table with one chip. If you aren't willing to lose it then why did you buy it? It could be the moment that your luck turns around.