Craps and Betting Systems

Date: 03/17/2003 at 11:10:28
From: Cliff Thomas
Subject: The game of Craps

In craps, can I bet on the 7 and win (more than I lose) if I assume a 
seven normally comes up within 32 rolls?

I plan to only start betting after I have not seen a seven for 5 rolls 
so I start betting on roll #6. (I know that the past doesn't influence 
future dice rolls.) Then I gradually increase my bets to ensure a 
profit of about $10 when I finally hit. The confusing part is whether 
over the long term I can win my bankroll amount before losing it all 
to when that stretch of no 7 extends beyond 32 rolls. And the fact 
that I am not betting for 5 rolls after a 7, which means when a 7 
comes up sometimes and I am neither winning or losing.

Not being a gambler I found that if I hop the 7 in bets of multiples 
of $3, I could increase my returns and allow myself to extend my rolls 
to 34 rolls still using my $6k bankroll. I am still trying to find 
math that can clue me in as to whether I am likely to win my bankroll 
back before losing it all to a streak of no 7 for 34 or more rolls. Is 
there a formula to help me figure this out?

Date: 04/11/2003 at 13:20:31
From: Doctor Achilles
Subject: Re: The game of Craps

Hi Cliff,

Thanks for writing to Dr. Math.

There is a house edge on every bet in the casino. Craps is actually 
one of the games with the smallest house edge, if you stick to the 
pass line and lay odds on every bet. Laying odds on your pass and 
come bets is actually the only place in the casino where you can make 
a bet where there is no house edge. In other words, the payoff on 
laying odds is exactly equal to the probability that the outcome will 
occur.

My personal favorite site for gambling rules and advice is

   The Wizard of Odds
   http://www.wizardofodds.com/ 

The author of the site is extremely methodical and mathematical and 
he does an outstanding job of explaining in mathematically sound ways 
how each game works and the pros and cons of different strategies.

You can go directly to the craps page at:

   http://wizardofodds.com/games/craps/ 

The problem with doubling your bet (or multiplying it by 3, or by 
1.5, or using some other strategy of increasing your bet every time 
you lose) is something that you hinted at already: you stand to lose 
your entire bankroll for the potential of a small profit.

You can put a couple of thousand dollars on the line at a $10 craps 
table and double your bet on the "don't pass" line every time you 
lose. If you start with, say, $1000 and you double your bet on the 
don't pass line until you either win once (and come out $10 ahead) or 
lose all your money, then you have a little less than a 99% of coming 
out ahead $10. If you start with $10000 and do the same thing, then 
you have a little less than a 99.9% chance of winning.

If you turn that around, you have just over a 1% chance of losing in 
the $1000 case and just over a 0.1% chance of losing in the $10000 
case.

If you lose, then you are down either $1000 or $10000.  A little more 
than 1% multiplied by $1000 is equal to a little more than $10 and a 
little more than 0.1% multiplied by $10000 is equal to a little more 
than $10.

So, on most trials you will win $10, but you have a small chance of 
losing a whole lot of money. And, on average, you will lose a bit 
more than you win.

If you do this only once, then you will probably win $10, but you may 
be unlucky and lose it all.

If you keep trying this over and over, then your odds of winning every 
time drop significantly.

Let's say that you start with $1000 and play a $10 table, doubling 
your bet until you win, then you start over at $10 and repeat. If you 
keep doing this until you are either up to a total of $2000 or down to 
nothing, then the odds that you will win are EXACTLY the same as if 
you just sat down at a $1000 craps table and laid all your money down 
on the first shooter.

In other words, on average, you expect to lose just as much as anyone 
else, using any betting strategy (assuming you put the same amount of 
money on the table). You can find a more complete explanation of 
betting strategies and the problems with them back at the Wizard of 
Odds:

   http://wizardofodds.com/gambling/scam.html 

It sounds like a great idea to sit at home with an Excel spreadsheet 
and plan out betting strategies with which you increase your bet. I've 
done it more than once. I once took one of my betting strategies 
(which involved multiplying my bet) to a casino. At one point while I 
was doing the strategy, I ran into a run of bad luck and had to double 
my bet 6 times in a row on a $5. All of a sudden, I had over $300 
riding on a $5 table. At that point, things didn't look quite as good 
as they did on my Excel spreadsheet. Because the dice don't remember 
how many times you've lost, you're no more likely to win after 
doubling your bet 6 or 7 times than you are the first time. So here I 
was with over $300 bet, and slightly less than a 50% chance of ending 
$5 ahead of where I started and slightly more than a 50% chance of 
ending several hundred dollars lower than where I started.  
Fortunately, I won that bet (and walked away with my $5 profit), but 
the experience taught me that betting systems are the sort of thing 
that look nice on paper, and are fine before you start using them. If 
you're lucky and conservative, they can be fun and (with enough luck) 
even profitable. But increasing your bet when you're down will 
occasionally leave you in a position where you either have to give up 
or bet more money than you want to lose so that you can have a few 
more dollars when you leave the casino than when you entered it.

Hope this helps.  If you have other questions or you'd like to talk 
about this some more, please write back.

- Doctor Achilles, The Math Forum
  http://mathforum.org/dr.math/