Video poker’s risk of ruin formula

For games with widely ranging payoffs and probabilities, such as video poker, it has been very difficult to determine bankroll requirements. Until recently, we relied mostly on computer simulations for an approximation, but Russian mathematician Evgeny Sorokin has shown that the risk of ruin (the probability of losing one’s entire starting bankroll) can be calculated precisely as a root of equation 1 below.

The formula may look complicated, but don’t let that stop you from reading this article, even if you’re not mathematically inclined. As a serious player, you will find this information very valuable. For several of the most attractive games, you can skip the math altogether by using a table published in Video Poker Times. Take a few minutes to look at the equation.

Simply stated, the risk of losing a one-bet bankroll is equal to the summation of a function of all the possible payoffs and their probabilities. Although the rebate is not actually paid immediately on each play as implied by adding C to W, the result is very close, since you can go to the slot club booth and make the cash any time it reaches the club’s minimum transaction level. As a starting point, this formula assumes that your entire gambling bankroll consists of exactly one betting unit (five coins for a typical video poker game), but it’s easy to extend that to a real bankroll. You place your one bet, and if you lose, you’re done. If you win or get a push, you continue playing indefinitely until you either lose that starting bet and all winnings, or have amassed a fortune.

Let’s apply this to a specific game. Suppose you play full pay double bonus poker very accurately in a casino that pays 0.25% cash rebate on the slot card, and you want to know your risk of ruin (RoR). The per-coin payoff (assuming five coins played) and the probability of each final hand type (assuming perfect play) are shown in the double bonus poker strategy section, and C will be 0.0025 for all cases. The Zilch (no payoff) entry is never shown on the machine’s payoff schedule, but its probability is given by the game analysis program, and it must be included in this calculation as the i=0 case.

Since R(1) is the summation of a series involving a function of itself, it can be found only by an iterative procedure; that is, make a guess, try it, use the result as the new guess, and repeat until the result is as accurate as desired. The series will converge faster if we start with a reasonable guess such as 0.99 and then average each result with the previous guess for use as the next guess. Using final hand probabilities and payoffs for standard full pay double bonus poker plus 0.25% slot club rebate, we find that the risk of losing a one-unit bankroll is R(1) = 0.99970164. (With real human play, the risk is a bit higher, typically in the fifth or sixth decimal place with skillful play)

Subtracting from one, we see that the probability of building a single 5-coin bet into a vast fortune with perfect play on this game is 0.0002984, or about one chance in 3,350. That may not seem like a very good prospect, yet it’s a lot better than any state lottery. Also, your starting bankroll is presumably at least a little bigger than just one 5-coin bet. What we really want to know is the risk of losing that real gambling bankroll. The probability of losing an entire starting bankroll of B betting units is given by equation 2. Since R(1) is less than one for a positive expectation game, a bigger starting bankroll results in an exponential reduction in the risk as follows:

For example, if you start out with 1,000 betting units (e.g., $5,000 on a 5-coin $1 double bonus machine) and play at a casino with a 0.25% cash rebate slot club until you either lose it all or amass a vast fortune, you have almost a 26% chance of becoming very rich (although at an expected average win rate of only 0.4% of your action, it’s going to take a very long time).

As a more practical concern, you probably want to know how big a starting bankroll you need for a specific risk of ruin that you consider acceptable. For example, suppose you decide that you can live with one chance in ten of losing your entire bankroll. You could interpolate from the above table and estimate that you need about 8,000 betting units for RoR = 0.1, but why not calculate it exactly? Solving equation 2 for B, we get equation 3. Plugging in 0.1 for R(B) and 0.99970164 as determined above for R(1) we get B = 7,715 betting units. Multiplying by $1.25, we find that we need to start with at least $9,644 to limit our RoR on a quarter Double Bonus game to ten percent.

Now let’s look at another game. For standard full pay deuces wild, R(1) computes to 0.9993469 (with no slot club rebate). This doesn’t seem much different from the figure for double bonus poker, but the third column above shows what happens when we raise it to large powers of B to compute RoR. Note how much lower the risk is with any reasonable bankroll. Combine that with up to five times the average win rate and a much more straightforward strategy, and you see why I recommend deuces wild for beginners in Las Vegas. On a quarter deuces wild we would need only $4,406 for that same 10% RoR, and that is even without a slot club rebate!

As you can see, the risk of ruin when playing video poker can now be determined precisely. You can do the calculation yourself, or Video Poker Times issue 7.5 presents a table giving the necessary bankroll for whatever you consider to be an acceptable risk of ruin for a dozen attractive games. (Note: There is a typographical error in the cash rebate for Sam’s Town in that table. It should say 0.093%. The RoR numbers for bonus deuces are correct for that rebate.)

There are always two roots of Sorokin’s equation; one, and the result we are seeking, which we are hoping is less than one. (If the root is greater than one then the game offers less than 100% total payback, and ruin is certain if you play long enough.)

equation 1:
equation 2:
equation 3:

Key: R(1) is the risk of ruin with a one-unit starting bankroll, n is the number of entries in the game’s payoff table, i identifies one entry in the payoff table, P is the probability of final hand type number i, W is the per-unit-bet win (payoff) for hand type i, C is the cash rebate from the slot club as a decimal fraction, B is the number of betting units in your starting bankroll, and R(B) is the risk of ruin with a starting bankroll of B bets. Note: One betting unit is typically five coins for video poker.