Transcript of the Gambler’s Ruin Problem Podcast
Dave talks about the Gambler’s Ruin Problem. It is a real equation for those who do not know. Not only is he recommending the book from which he first learned of the Gambler’s Ruin Problem but he’s sharing his experience and teaching on it!
My primary article today is about the gambler’s ruin problem. I want to tell you about this book written by Alan Wilson, “The Casino Gambler’s Guide,” which was published in 1965 by Hart Munroe. This is a 5-star book for anyone who takes gambling seriously.
It covers everything from dice to bias to roulette wheels to blackjack and more. I highly recommend this book and suggest you acquire it if you can find a copy. I am currently out of copies of the book except for my own original edition. If you search for it on the web you’ll find that you can probably get it for somewhere between $30 and $40.
If you want something better than that, I’ve seen deluxe issues of this book retailing for $125 and more. I strongly suggest that you buy this book. But, don’t be confused. There are other, similar books out there that are not as good. So make sure you purchase “The Casino Gambler’s Guide” written by Allen M. Wilson.
I am going to tell you a story I first told in 1991 at a seminar which, I believe, took place in Reno, but might have actually occurred somewhere else. I’ve told this tale many times. There is a punch line here because I actually got some of it wrong.
The story illustrates what gambler’s ruin is all about. The gambler’s ruin problem essentially covers many forms of gambling, but we are going to apply it specifically to flat bets on the roulette wheel. Gambler’s ruin addresses the possibility of winning a session of x number of units. As I’ve already noted, we are going to use roulette as an example.
I want you to consider that when I first told this story you could actually bet $1 on the roulette wheel all over Las Vegas. Now I think you would be hard-pressed to even bet as little as $5 or $10, maybe even more than that.
Now imagine that there is a guy who comes to Las Vegas every weekend to play roulette and he comes with precisely $100 to gamble. He bets $1 at a time on the color red at the roulette wheel. The math of the roulette wheel is not complicated if you bet on red. There are thirty six primary numbers plus zero and double zero on the wheel. That means there are a total of eighteen reds, eighteen blacks and two greens.
You have eighteen chances out of thirty eight and the disadvantage in that game is 5.26% if you do the math. The question is this: does this man who bets $1 at a time, from a starting bankroll of $100, have a reasonable chance to win? He is likely to continue playing until he doubles his stake or until he runs out of money.
In other words, he is trying to win 100 units before he loses 100 units. I presented this question for the first time to a seminar group. At that time, I said that if the man visits a casino once a week with a stake of $100, intent on playing until he wins or loses $100, how often does he win?
The first time I told this story one fellow raised his hand and said,“I am going to apply a little logic here.” If the gambler has a 5% disadvantage that would mean his win-loss ratio would be 47.5%-52.5%. If he visits a casino once a week, and there are fifty two weeks in the year, then I would suggest he would win about twenty four times, a 47% win rate.
Another guy put forth an opinion, too. He stood up and said: “No, no, no, there is a compound impact here. It is harder than that. I think it is more like 22 wins a year.”
You can make your own mental guess. What are the chances of such a player winning? The whole idea is to win 100 units before you lose 100 units while never betting more than one unit at a time. It is a flat bet and you have a 5.26% disadvantage.
The funny thing is that these two guys in the audience argued vigorously, their voices loud and angry to the point where we really thought someone was going to have to intervene and calm them down. I said to them that the answer to the question posed would leave them so astounded that they would quickly realize how ridiculous it was for them to quibble over it.
Interestingly, I then gave them an incorrect answer. I know that because I recently reworked the calculations and discovered that, although I have been telling this story for over 20 years, I have been telling it wrong. The answer that I gave the two men was that the gambler would win once in every 640 years!
The punch line is that I was wrong. It is not 640 years. It is actually 6,410 years!
What does this have to do with anything? I am going to spend a moment on the calculation to help you to fully understand the impact.
Think of it this way. If you went to Las Vegas with that $100 bankroll and broke it into a single, $100 unit, you’d make just one bet, and, win or lose, be done for the week. In that case, if you went 52 times in the year it is logical that you would win that single bet about 47% of the time.
Now, if you used two $50 units or four $25 units, the math would change. Can you imagine that the more units in your session, the more difficult it is to get ahead by that number of units? In other words, it is harder to win five units than it is to win one unit.
Further imagine that the more units in the session, the more difficult it is to overcome the mathematical disadvantage – the takeout -of 5.26%.
Consider the following: with one unit you have a 47.37% chance of winning. If you went to five units and took that same $100, breaking it into $20 units in order to win $100, you would have to finish ahead by a total of five units.
So, going from one unit to five units has reduced your chances of winning substantially, from 47% to 37%!
If you go to ten units – so you’re now betting $10 at a time – you have to get ahead by ten units. This drops your chances of winning to 26%!
It gets even worse. At 25 units you have only a 6.7% chance of winning.
In other words, if you go with 25 units, betting just $4 at a time and the goal is still to win $100, you have greatly reduced your chances of getting lucky. The truth is that you’ll win roughly once in every 15 trips!
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