Gambler's ruin
From Wikipedia, the free encyclopedia
The basic meaning of gambler's ruin is a gambler's loss of the last of his bank of gambling money and consequent inability to continue gambling. In probability theory, the term sometimes refers to the fact that a gambler will almost certainly go broke in the long run against an opponent with much more money, even if the opponent's advantage on each turn is small or zero.
Contents |
[edit] Examples
[edit] Coin flipping
Consider a coin-flipping game with two players where each player has a 50% chance of winning with each flip of the coin. After each flip of the coin the loser transfers one penny to the winner. The game ends when one player has all the pennies. If there are no other limitations on the number of flips, the probability that the game will eventually end this way is 100%.
How do you calculate this certainty?
If player one has n1 pennies and player two n2 pennies, the chances P1 and P2 that players one and two, respectively, will end penniless are:
Two examples of this are if one player has more pennies than the other; and if both players have the same number of pennies. In the first case say player one (P1) has 8 pennies and player two (P2) were to have 5 pennies then the probability of each losing is:
= 
= 0.3846 or 38.46%
= 
= 0.6154 or 61.54%
It follows that even with equal odds of winning the player that starts with fewer pennies is more likely to fail.
In the second case where both players have the same number of pennies (in this case 6) the likelihood of each losing is:
= 
= 
= 0.5
= = =0.5
In the event of an unfair coin, where player one wins each toss with probability p, and player two wins with probability q = 1-p < p, then the probability of each winning is:
This can be shown as follows. Consider the probability of player 1 experiencing gamblers ruin having started with n > 1 amount of money, P(Rn). Then, using the Law of Total Probability, we have
,
where W denotes the event that player 1 wins the first bet. Then clearly P(W) = p and 
. Also P(Rn | W) is the probability that player 1 experiences gambler's ruin having started with n + 1 amount of money: P(Rn + 1); and 
is the probability that player 1 experiences gambler's ruin having started with n − 1 amount of money: P(Rn − 1).
Denoting qn = P(Rn), we get the linear homogenous recurrence relation
- qn = qn + 1p + qn − 1q,
which we can solve using the fact that q0 = 1 (i.e. the probability of gambler's ruin given that player 1 starts with no money is 1), and 
(i.e. the probability of gambler's ruin given that player 1 starts with all the money is 0).
[edit] Casino games
A typical casino game has a slight house advantage, which is the long-run expectation, most often expressed as a percentage of the amount wagered. In most games, this edge remains constant from one play to the next (blackjack being one notable exception).
Calculating House Edge
If the game played has an actual pay off of w and true odds of t then the house edge can be calculated from the formula:
For example, if you have true odds of 15 to 1 and a payoff of 14 to 1 then the house advantage is:
= 
= 0.0625 = 6.25%
For example, the official house advantage for a casino game is 6.25%, and thus the expected value of return for the gambler is 93.75% of the total capital wagered. However, this calculation would be exactly true only if the gambler never re-wagered the proceeds of a winning bet. Thus after gambling 100 dollars (called "action") the idealized average gambler would be left with 93.75 dollars in his bankroll. If he continued to bet (using his 93.75 dollars in proceeds as his new bankroll), he would again lose 6.25% of his action on average and the expected value of his bankroll would go down to 87.89 dollars. If the proceeds are continually re-wagered, this downward spiral continues until the gambler's expected value approaches zero. Gambler's ruin would occur the first time the bankroll reaches exactly 0, which could occur earlier or later but must occur eventually.
The long-run expectation will not necessarily be the result experienced by any particular gambler. The gambler who plays for a finite period of time may finish with a net win, despite the house advantage, or may go broke much more quickly than the mathematical average.
Gambler's ruin can also occur if the player has an advantage over the house, as in Blackjack card counting, because the player has a limited bankroll.[1]
[edit] Sports Betting
Sports betting in America is based on money line odds, or numbers on a standard number line moving from negative to positive values. Negative values are points that go to the favorite, and positive values are points that go to the underdog. Its important to understand that your total winnings, is equal to your initial wager + your winning amount. (Total winnings=initial wager + winning amount)
An example of the (+), (-) lines are:
Red Sox -120
Rockies +180
This means that you wager $120 on the red sox to win $100, and you’d wager $100 to win $180 on the Rockies.
Other types of odds that are seen around the globe are Fractional and Decimal odds.
Fractional odds are mainly used in the UK and Ireland. An example of fractional odds is 
, read “four to one”, these are odds against you meaning you’d wager $400 on a $100 stake. And vice versa if the odds are 
, read “one to four”, these are the odds in your favor, meaning you’d wager $25 on a $100 stake. Fractional odds of 1/1 are even money odds, you’d wager $100 to win $100.
Decimal odds are very similar to fractional odds except in this case you have to part with your wager first in order to win. The values for decimal odds are the values for fractional odds + 1. (i.e. =0.25 + 1 = 1.25). These particular types of odds are ideal for trading stocks and parlays in sports betting.
[edit] Notes
[edit] See also
[edit] References
Epstein, R. The Theory of Gambling and Statistical Logic. Academic Press; Revised edition (March 10, 1995)
