How Much You Win When You Win
On Craps, Ergodicity and Why You Should Have Left the Casino an Hour Ago
There’s an old saying around the casino that goes: “It’s not about how much you win, it’s about how much you win when you win.”
I have always dismissed it when I heard it. Of course the casino would want me to focus on my winnings while picking my pockets with negative expected value games, but as often happens with old sayings, it speaks to a deeper truth about gambling. Let me explain:
It’s a night out in Vegas and you are playing craps. There’s a ton of shouting and yelling and you’re not quite sure what’s going on. Little did you know, you’ve picked the only game in the casino that contains a bet with no house edge at all.
Craps is played by throwing a pair of dice across an oblong table. Bets are made on the outcomes of a series of rolls by a single shooter (for odds and edges check out the graphic). The series ends when a shooter craps out. There are two phases of play. In the first, the shooter is “coming out” or rolling to establish a point. Before the come-out roll, you place a pass line bet on the table. Think of it as betting with the shooter (we wont talk about the don’t pass and equivalent backing bet—don’t bet against the shooter). If they roll a 7 or 11 in this phase, pass line bets pay and the shooter keeps rolling. Roll a 2, 3, or 12 (the “craps” numbers) and the pass line loses immediately. If another number is rolled it becomes the point and we move to the second phase, where we get access to the Free Odds bet—the best deal in the casino.
CRAPS — THE FULL BRIEFING
The free odds bet pays according to actual probability. This is unusual. Almost every other bet in the building pays less than true odds and that gap or ”edge” is how casinos make money. Free odds has no edge at all. You get 2:1 odds on points 4 & 10, 3:2 on 5 & 9, and 6:5 on 6 & 8. (see graphic) Because free odds is such a deal, most casinos will limit how much you can wager to 10x your pass line bet.
If this was a totally free lunch casinos would never allow it, so lets consider the straight math for just a moment. You only get access to the odds bet if you have a pass line bet on the table which has a house edge of ~1.4%. Your max odds bet cuts that edge down on the combined bet by a factor of 10, so the edge falls to around 0.14%.
Some sharper readers may now be shouting at the screen: “What did that help? There’s still a house edge.” You’re right. Let’s say you roll 30 times per hour—a reasonable average—and play for 5 hours starting at $1,000. Your expected value is $979 walking home. A $21 loss on a $1,000 night out. Honestly, not terrible. But expected value doesn’t tell the whole story, because EV is a statement about averages and you, as a single embodied player moving linearly through time, are not and could never be average.
Let’s back up
When analyzing probabilities through time there is a concept called ergodicity. The distinction it draws is between two different kinds of average. The ensemble average (or Expected Value) asks: if a thousand different people each played one session tonight, what would their average outcome be? The time average asks something different: if you played a thousand sessions over your lifetime, what would your average be? In a well-behaved system those two numbers are the same. Gambling, it turns out, is not a well-behaved system.
Why is the game of craps so unruly? Because its outcomes include the risk of ruin. Expected value/ensemble averages assume you can always make the next bet. If you go broke in hour two, that assumption is false—and no amount of good expected value in hours three, four and five can save you, because you are no longer playing. To see how often this actually happens we’ll need to run a simulation. Shall we say 1,000 bettors betting 30 times an hour for 5 hours? Good. Love that.
STOP-WIN COMPARISON
Now we can see that while the ensemble averages converge around $970 or so, the most likely single outcome actually is going broke with between 25%-40% hitting bust depending on the run. (the simulation runs 1000 players and then graphs 200 of them, so keep refreshing the sim for a better idea of possible outcomes). Look at those paths, they are incredibly volatile. Exhilarating runs followed by gut wrenching slides. This is the romance of the craps table and it has everything to do with the relationship between your bet size and the house edge. The house takes $0.14 per round in expectation, but we are betting $110 on each point, which means the standard deviation per round is something like $108. With fluctuations like that going broke becomes ever more likely, but hold on—what if there was a way to tame this volatility and use it to our advantage?
If there was only a way to choose which part of the distribution we got to experience, perhaps we could tilt the odds in our favor.
Enter the “Stop Win”
A stop-win is a rule designed to make you cash out once a specific bankroll is reached. You decide it before you walk in the door. Starting with $1000, lets say once you hit $1,500 you’re done, cash out. Remember, it’s not about how much you win, it’s about how much you walk away with when you win.
What happens to your traditional odds when you put in a stop-win? They go down. Let’s say you put a stop-win at $1,500—your expected value at the end of 5 hours of play dips by about $1. Move it to $2,000 and it falls by around $35. This sounds like bad news right? but hear me out. Expected value falling slightly just means you have surrendered the rare sessions that would have run past your target and kept climbing—the unicorn sessions that the mean was quietly averaging in. You are trading a small slice of upside that you were unlikely to experience anyway for something much more valuable: a controlled distribution.
Because you are not the ensemble outcome. You are one person, on one night, living one path through the simulation. On the night ~60% of bettors hit a $1,500 stop-win and end their night up $500, 34% of bettors lost it all and the remainders limped along. The stop-win controls the median outcome, taming volatility and resulting in a decent chance you walk out with $500 in profit.
It’s important to highlight that a stop win doesn’t negate the house edge—it just makes it fairly likely that you walk away a winner. The casino knows this. They are counting on the gap between your peak and your walk-away—every round you play past your high watermark is a round the edge works against a player who has already, in some sense, won. The stop-win closes that gap by force, but most of us of course, won’t use one. We’ll sidle back up to the table for one more throw of the dice, the table’s hot, and we’re feeling lucky after all.
P.S.
There are some fun graphics on this post, courtesy of the excellent data visualization capabilities of Claude.ai which also helped edit this piece and provided the simulation data—its pretty cool to be able to simulate a couple thousand throws of the dice just by asking, give it a try if you haven’t. This post started out as a simple conversation on the distribution of dice throws and built into something I think is genuinely interesting—technology can be pretty cool sometimes.
P.P.S
I am not a statistician and this information is the result of one conversation with AI, if you have expertise in the field and want to suggest some edits please do so in the comments! I’d love to learn more
Here is the final interactive guide—it shows the optimal stop win number based on a series of factors. Interestingly enough I figured the optimal number for winnings would be the point where the bust rate and hit stop win rate cross. Actually slightly further back on the curve is optimal as the bust rate grows nonlinearly the longer you play. Play around with the sims, they are small enough sample sizes to have the numbers shift substantially when re run. This highlights how the extreme volatility of outcomes in Craps (when you bet optimally) opens it up to some interesting strategies by players.