# The Gambler’s Ruin Explained – Fair Coin Flipping

One of the phenomenons of probability is Gambler’s Ruin. The most common meaning is that a gambler with finite wealth, playing a fair game (that is, each bet has expected zero value to both sides) will eventually go broke against an opponent with infinite wealth.

Photo: Dario Sabljak (Shutterstock)

In other words, the maxim of gambler’s ruin is that if you play long enough you will eventually go bankrupt and have to quit the game prematurely. The common gambler has far less money than a bookmaker or casino and there will inevitably be a time when he will simply be unable to continue playing and, of course, the house will not be giving credit.

“Long enough” may be a very long time though. It mainly depends on how much money the gambler starts with, how much he bets, and the odds of the game. Even with better than even odds, the gambler will eventually go bankrupt. But, this may take a very long time indeed.

Please note that we are talking here about a “fair” game; e.g. each bet with zero value. The practice of bookmakers to offer odds with an overround in their favour makes this outcome just much quicker.

### Fair Coin Flipping

To make the dilemma of gambler’s ruin a little easier to understand imagine coin flipping with a friend. You each have a finite number of pennies (n1 for yourself and n2 for your friend).

Now, flip one of the pennies (either player). Each player has a 50% probability of winning (head or tail). If it’s a head you win a penny and if it’s a tail you surrender a penny to your friend. Repeat the process until one of you has all the pennies.

If this process is repeated indefinitely, the probability that one of you will eventually lose all his pennies is 100%. In fact, the chances P1 and P2 that players one and two, respectively, will be rendered penniless are:

Now let’s populate these equations with real numbers:

The above example is based on both players starting with the same amount of pennies (100 each). In other words, you and your friend have both an exact probability of 50% to end up with all of the pennies after many, many coin flips. This means that after an unknown number of coin flips either you or your friend will finish banking all the pennies. At the start, your chances are equal, and it is impossible to say who may win.

However, if one of you has many more pennies than the other, say you start with 100, and your friend with 10,000, then your chance of finishing with all of the pennies (yours as well as your friend’s) is as low as 1%, whilst your friend’s chances are 99% to win this unequal match.

Last Update: 21 April 2013