The mathmatics Behind Gambling

Gambling is a game enjoyed by adults around the globe, and you may be tempted to try it. Don't. The house, or the host of the game, e.g. the casino, will always win in the long run. Let's look at why by looking at the card game craps!

The Rules of Craps

Craps starts with a player rolling two dice. A roll of 7 or 11 results in a win for the player. A roll of 2, 3, or 12 means the house wins. Rolling a 4, 5, 6, 8, 9, or 10 establishes your target number. You then roll again. If you hit your target number, you win! However, if you roll a 7, you lose. If your target number isn't rolled and you don't roll a 7, you keep rolling. This is continued until you win or lose.

The Math I: Winning/Losing on roll 1

The probability of getting a 7 is 6/36(1/6), as there are 6 ways to roll 7 using 2 six-sided die. The chance of getting an 11 is 2/36, so your chance of winning on your 1st roll is 8/36(2/9). Your chance of losing, however, is 1/36+2/36+1/36=4/36=1/9! So, you have a better chance of winning on your first roll than losing. But let's look at what happens if you don't win or lose on your 1st roll:

The Math II: Rolling a 4 or 10

You have a 3/36 chance of rolling a 4 on the first roll, same for 10. So, if you rolled a 4 or 10, you would have a 1/6(6/36) chance of losing in the very next round(roll a 7) and a 3/36 chance of winning next round. If you get any other roll, you end up in the same senario, so we can scale the fractions up to equal 1. so, you have a 2/3 chance of losing on a 4 and a 1/3 chance of winning.

The Math III: Rolling a 5 or 9

Same, except you have 4/36 chance of winning next roll. scale up to get 2/5 chance of a win and 3/5 chance for a loss.

The Math IV: Rolling 6 or 8.

Same, but you have a 5/36 chance of winning next roll, so scale up to get 5/11 win and 6/11 loss.

Final Calc: Math V

So, your chance of winning by rolling a 4 or 10 than using to get a point is 2*1/3*1/12=2/36=1/18.

For 5 or 9: 2*2/5*4/36=4/45

8 or 6: 2*5/11*5/36=50/396. the sum of those is 26/99, add your chance of winning on your 1st roll(2/9)=48/99! That is your fraction! Bye for now!