Craps Odds: The Casino Math Behind The Game

So we’ve talked about Roulette in the first article of this series. Now let’s move on to the next game where the mind-boggling odds I’m about to lay on you are sure to make you crap in your pants! They actually won’t, but I can’t resist word play. Sorry.

The Craps Table

The Craps table can seem a little intimidating to a casino newbie. There’s often lots of noise, frantic arm-waving, and generally an exciting (or tense) atmosphere. Moreover at first glance, figuring out the craps table can appear to be a daunting task. I remember the first time I visited a casino and came across one – I didn’t even dare approach it because it seemed packed with people that appeared to be ‘seasoned’ gamblers. Craps odds and the game isn’t all that complicated though.

I’m not going to go through the rules of the game here (because I’m assuming you already know them if you’re reading an article on the mathematical analysis of the game), but just in case you don’t, I find that this video explains it better than most. It’s ridiculously retro (you can tell it was likely uploaded from a VHS tape) but gets the job done!


Players enjoying craps odds

Clearly enjoying my “crap in your pants” joke more than you did.


The Numbers Behind Craps Odds

So let’s talk numbers.
We know that when we roll a 7 or 11, we win; a 2,3 or 12, we lose and everything else gets us rolling again. But what exactly are our chances of a win?

Well there are 36 two-dice roll possibilities in total. And If you thought that the probability of rolling every outcome (values 2-12) was similar, don’t feel too stupid. I’ve come across SO many people that were under the same impression. It’s very wrong though. The probabilities of each outcome are as follows:


Total Probability
2 1/36
3 2/36
4 3/36
5 4/36
6 5/36
7 6/36
8 5/36
9 4/36
10 3/36
11 2/36
12 1/36

As you can see, your odds of rolling ‘lucky number 7’ are much higher than any other. Right about now you’re beginning to wonder – “OMG….but I win when I roll a 7! Does the casino know their craps odds?!!”

Sid: LOL, yes they most certainly do.

To achieve a natural win you need to roll a 7 or 11 = 8/36 = 2 out of 9 times you win immediately
To achieve a natural loss you need to roll a 2,3 or 12 = 4/36 = 1 out of 9 times you lose immediately

Loving it so far, yes?

And rolling anything else gives you a ‘point’ = 24/36 = 2 out of 3 times you’ll need to roll again

Here’s where the casino stops playing nice and takes ‘lucky number 7’ away from you.
So now you need to keep rolling until you hit your ‘point number’ and win or hit a 7 and lose. Odds for this aren’t quite as lovely I’m afraid.


Craps Odds with Apu

Thank you, Come again.


To illustrate, when you roll a point number (let’s say 4) and have to keep rolling until another 4 appears and you win or a 7 appears and you lose, there are effectively a total of 9 outcomes that should matter to you:

Total outcomes = 36/36
Outcomes that don’t matter (ie. not a 4 or 7) = 27/36
Outcomes that matter (ie. 4 or 7) = 9/36

Now out of these 9 outcomes,
Rolling 4 = 3/9
Rolling 7 = 6/9

Anyway you calculate it, your chances of winning when you roll a point number is lower than your chances of losing.

If you go through the calculations when you roll a 5, you will realize your chances of winning will be 4/10. While better than rolling a 4, it’s still not going to beat 7 in the long run. Also, in case you were wondering (because you’re too lazy to calculate it yourself), rolling a 6 = 5/11 chance of winning.

Here’s an overview:

Opening Roll Winning Chances
2 0
3 0
4 3/9
5 4/10
6 5/11
7 1
8 5/11
9 4/10
10 3/9
11 1
12 0

Let’s go one step further using the Law of Probability to determine the exact winning odds for each roll. It states that the overall winning chance is a weighted average of your chances from each opening roll.

So when we weigh each opening roll based on how likely it is to occur on the first roll, the data will look like this:

Opening Roll Weight Winning Chances Overall Winning Chance
2 1/36 0 0
3 2/36 0 0
4 3/36 3/9 9/324 = 0.0278
5 4/36 4/10 16/360 = 0.0444
6 5/36 5/11 25/396 = 0.0631
7 6/36 1 6/36 = 0.1667
8 5/36 5/11 25/396 = 0.0631
9 4/36 4/10 16/360 = 0.0444
10 3/36 3/9 9/324 = 0.0278
11 2/36 1 2/36 = 0.0556
12 1/36 0 0
TOTAL 1 (or 36/36) 244/495 = 0.493

So all things considered, you have a 49.3% chance of winning at the game of Craps.
Hence, EV = 1(0.493) + (-1)(0.507) = -0.014

While it still works out to be a better proposition than Roulette, it clearly still doesn’t make for a sound long-term investment.


Hey! Take a gamble by connecting with us on social media. Odds are you’ll love it.

Images: mammaoca2008freakgirl