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Ask Pascal how to avoid going broke

May 6, 2008 7:00 PM

Keno Lil | The Gambler’s Ruin Problem was posed and answered by the mathematician Blaise Pascal several hundred years ago.

When a gambler faces a house possessing a large bankroll and having limited funds, he must eventually go broke. The odds of such a gambler of "breaking the bank" are vanishingly small. It is a principle all gamblers should keep in mind. The larger bank roll will win in the end.

I decided to test the problem using a Monte Carlo simulation on my PC. I arranged for 100 players, each with an initial bank roll of $10 to flip a coin versus the house. Heads, the player wins a dollar, tails, the house wins a dollar. It’s a 50-50 proposition, a fair game, with the only difference being the house has an unlimited (infinite) bankroll.

Not surprisingly, all 100 players lost their $10. This result does seem to fly in the face of common sense. In a fair game, some should win and lose. Given the random coin flips, shouldn’t some players lose more than they win (and go broke) while others win more than they lose and eventually break the bank?

Time is the problem. Every player will eventually have a long enough losing streak to bust out a limited bankroll. True enough, the house will also have long losing streaks, but given its much larger bankroll, it is far more able to ride them out.

In my simulation, it took over 5 million coin flips to bust out the 100 players, 5,285,132 to be exact. After many decisions, the house won $1,000. This was the result of a variation from the expected house win of less than 1 percent!

 The average player lasted 52,851 coin tosses before going broke. The median player lasted only 937 tosses before going broke. The luckiest player played 274,261 tosses before eventually going broke and achieved a net win of $1,155.00 before losing it all back again.

In this case, the house could have had a bankroll of only $1,200.00 and still withstood the efforts of all 100 players!

In a test where the odds are 50-50, the chances of a player going broke with a $10 bankroll and shooting for $1,000 are 99 percent. Only one out of the 100 players reached the $1,000 milestone before going broke.

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