Vegas bankroll rule -- you will always bust before casino

Oct 7, 2008 5:00 PM

Winning Strategies by Elliot Frome |

Last week, I introduced my own law of math, for which I borrowed the name of an existing law of math that is the law of large numbers.

To a mathematician, this law (the real one) says we take a random sampling of a number that is supposed to approach a finite value. The more samples we take, the more likely the average will approach this value.

In simpler terms, the more hands we play, the closer we come to the theoretical payback of the game.

My own version is if you start with your normal bankroll playing even a 100 percent payback game, the casino still has a tremendous (if not infinite) advantage because it starts with a significantly larger bankroll. This law basically tells us you’re going broke before the casino is, even if the game gives no one an advantage.

It’s like this. You and I flip a coin. If it’s heads I win $1 from you. If it’s tails, you win $1 from me. If we each start with $20, over time, we will each bankrupt the other about the same amount of times. If, however, I start with $20 and you start with $40, then I will go bankrupt twice as often as you.

If you’re playing an infinite number of sessions, then it really doesn’t matter. The game is still a 100 percent and, in the end, we’re each going to finish with the same amount we risked in total.

But the real world doesn’t work this way.

If I show up with $20 and you have $40, there is a reason. Generally, it is because $20 is all that I have to risk. As a result, I am really at a disadvantage. At the point I am down $20, I am done. At the point you are down $20, there is still an opportunity to turn the tables since there’s $20 left in your bankroll.

Fundamentally, the payback of the game is still the same, but the likelihood that I will run out of money first is double your risk. In math, this is called the risk of ruin, because essentially you are bankrupt at that point and there is no coming back.

Obviously, very few people really put all their money on the line and truly are ruined. However, they do reach the point at which they are not willing to put up more money for the time being.

Players show up at casinos with all different size bankrolls depending on what game they are playing and what denomination they are playing. But, I think it is more than fair to say that a good-size bankroll for someone in Las Vegas would be about $1,000. I’m sure many of you reading this probably head to the casino with far less. Some go with considerably more.

It’s not for publication, but I think it is safe to say that a casino is probably bankrolled for $100 million. Now, technically, this bankroll has to be used to play against every patron at the same time, but we have to simplify things a little. So, a casino has a bankroll that is about 100,000 times larger than a player.

So, even if we’re just flipping a coin, we’ll find that the player will go bankrupt 100,000 times for each time the casino does. Consider that most casino games play at less than 100 percent and limits the size of wagers. In some cases, they limit even the aggregate payout on any one hand. You can quickly see why the odds of ‘breaking the bank’ is in the range of astronomical.

From this we learn that a goal of gambling must be to set realistic goals in terms of when to quit when you are ahead. In the end, you are battling both the payback and the casino’s very large bankroll.