Gambler's Ruin

The term gambler's ruin is used for a number of related statistical ideas:

• The original meaning is that a gambler who raises his bet to a fixed fraction of bankroll when he wins, but does not reduce it when he loses, will eventually go broke, even if he has a positive expected value on each bet.
• Another common meaning is that a gambler with finite wealth, playing a fair game (that is, each bet has expected value zero to both sides) will eventually go broke against an opponent with infinite wealth.
• The result above is a corollary of a general theorem by Christiaan Huygens which is also known as gambler's ruin. That theorem shows how to compute the probability of each player winning a series of bets that continues until one's entire initial stake is lost, given the initial stakes of the two players and the constant probability of winning. This is the oldest mathematical idea that goes by the name gambler's ruin, but not the first idea to which the name was applied.
• The most common use of the term today is for the unsurprising idea that a gambler playing a negative expected value game will eventually go broke, regardless of betting system. This is another corollary to Huygens' result.

While the first three meanings have some relevance for gamblers, they are also general theorems with wide application and many related results in probability and statistics. Huygens' result in particular led to important advances in the mathematical theory of probability.