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Risk of Ruin
Video Poker for Fun and Profit
by Dan Paymar

Russian mathematician Evgeny Sorokin has developed a formula to precisely determine the risk of ruin in games with widely ranging payoffs, such as video poker. In this column, we will use the formula to examine a popular video poker game that offers over 100% payback. Sorokin shows that the risk of ruin (the probability of losing one’s entire starting bankroll), when that bankroll consists of exactly one betting unit, is a root of the equation:

R(1) = the Risk of Ruin with a one-unit starting bankroll,
n = the number of entries in the game’s payoff table,
P(subscript i) = the probability of final hand type number i, and
W(subscript i) = the per-unit-bet win (payoff) for hand type i.

Since the result we’re after, R(1), also appears in the formula itself, it can be found only by an iterative process. Also, a slot club cash rebate can significantly reduce your Risk of Ruin, and this can be taken into account by adding the rebate as a decimal fraction to each W in the above formula. (For example, if a particular payoff is 5-for-1 and there is a 0.25% cash rebate from the slot club, then use 5.0025 for that W.)

Now let’s look at a specific game. Suppose you play full pay (10/7) Double Bonus Poker very accurately, and you want to know your Risk of Ruin (RoR). A game analysis program tells us that, with perfect play, the expected long-term payback is 100.17%, and it gives the probability of each final hand. Using those probabilities in the above formula, we find that R(1)=0.9997674. That is, the Risk of Ruin is very close to certainty, yet if we subtract that from one to get the probability of not going broke, we find that there’s about one chance in 4,300 of building a single five-coin bet into a vast fortune on this game. That may not seem to be a very good prospect, but consider that ruin is certain with extended play on any game that returns less than 100%. But more to the point, your starting bankroll is hopefully much bigger than just one betting unit. The risk of losing an entire starting bankroll of (B) betting units is simply R(1) raised to the (B) power. Since R(1) is always less than one for a positive expectation game, a bigger starting bankroll results in an exponential reduction in RoR as follows:

Starting Bankroll Risk of Ruin
Betting Units Playing Forever
100 97.70
500 89.02
1000 79.24
3000 49.76
5000 31.25
7000 19.62
10,000 9.77

For example, if you start out with 1,000 betting units (e.g., $5,000 on a 5-coin dollar machine) and play accurately on 10/7 Double Bonus Poker until you either lose it all or amass a vast fortune, you have a 21% chance of becoming very rich. At an expected average win rate of perhaps ten dollars per hour on a dollar machine, however, it’s likely to take a very long time, so we play this game mostly for recreation but with a much better chance of ending up a winner than on a slot machine or any other negative expectation game. More discussion on Risk of Ruin analysis and the results for several popular games have been given in various issues of my newsletter, Video Poker Times.

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