# Discount Gambling

## Session Outcome Distributions for 5x Card Craps

Posted in card craps by stephenhow on August 28, 2010

I wanted to know what kind of variance to expect during my card craps sessions playing 5x Don’t Pass odds with a 3 roll window. So I calculated the cumulative distributions for a few strategies, based on a 100 bet bankroll (e.g., \$500 @ \$5 min Don’t Pass bet), a +100 bet goal (quit if double up), and a 200 game session limit (might take hours). I used simulations to find the probability density function for a single game (hand) of craps, then convolved it the number of session hands (given bankroll and goal) to get the final session distribution. Finally, I plotted the cumulative distribution function, since it’s easier to read for gambling purposes.

Card Craps Session Outcome Distributions

First, I compared regular dice to card craps, both laying 5x DP odds every roll. As the two curves show, they’re very similar, although the card craps game (magenta, -0.1%) returns better than normal dice (red, -1.36%). In both cases, you have more than a 20% chance of busting out (e.g., -\$500 @ \$5 pass line bet) or doubling up. (If you closely compare the dice and card curves, you’ll see the cards have almost an equal chance of busting out or doubling up, while the dice show a bias towards busting out.)

Next, I found that card craps performs equally well laying 5x odds with a 3-roll window or a 6-roll window. This is a nice find, since it’s pretty easy to implement the 3-roll windowed count, but practically impossible to count with a 6-roll window. [If you look closely, the 6-roll window (green) performs a little better than the 3-roll window (blue)]. The session distribution function shows about a 12% chance of busting out or doubling up. If you see where the cumulative distributions cross 50%, you’ll see that the card craps games are slightly +EV, and the dice game is of course -EV.

Not surprisingly, playing 5x DP odds with card craps has less variance than actual dice. This results mostly from the lower average bet (4.56 units/roll) for a 3-roll window, compared to dice (6.37 units/roll).

Having calculated this cumulative distribution function, I’ll probably play 5x DP with a 3-roll count, and grind out the player reward points. That’s probably the best reason to play, because you’ll easily make the Elite membership level, with it’s \$20/day Free Play award. You’ll also probably make more in reward points (1 point = 1 cent) at the game than in +EV (maybe \$0.005/roll). Of course, there’s always the satisfaction of beating the house, whatever that’s worth to you.