There's no easier way to put the average person to sleep than to start talking about the standard deviation of a casino game. Except, perhaps, to start talking about variance. Wait! Wake up!

We're all interested in winning, right? When we set out to play several hours of blackjack or video poker, we hope we'll end up with more money than we started with.

Most of us intuitively think that the lower the house edge, the more likely we are to win. A blackjack game that pays 3:2 for a natural is a whole lot better than one that pays 6:5. A Jacks or Better ("JOB") video poker game that pays 9 for a full house and 6 for a flush is way better than one that pays 8 and 5. But the game with the lower house edge is not necessarily the game you are more likely to win at.

For example, if you play one hour (say 500 hands) of 99.94% Super Aces video poker, you will have a 29% chance of winning. You'd have a greater chance of winning-38%--if you played one hour of 99.54% JOB, even though that game has a bigger house edge.

So why is it, in the scenario above, that you are more likely to lose when playing the game with the smaller house edge? It's because Super Aces has much larger payoffs for relatively rare hands, especially for four Aces (but also all other quads). In contrast, Jacks or Better has larger payoffs for more common hands, especially two pair (but also full house and flush). With Super Aces, you'll average one of the larger payoffs every 400 hands, but you can easily go 500-1000 hands without hitting a single one. Meanwhile, the JOB player is collecting extra money (compared to Super Aces) every four or five hands, because that's how often two pair, flushes, and full houses occur.

"Variance" is a way of measuring the difference between games that depend heavily on rarer hands and games that don't. If we know the variance of one hand of a particular video poker game, we can easily calculate the variance of any number of hands. That's because the variance of two hands of video poker is just twice the variance of one hand. And that's true for any number of hands. You can just multiply the single-hand variance by the number of hands to get the total variance. That may seem trivial, but it's not. For example, the standard deviation of two hands is NOT twice the standard deviation of a single hand. Perhaps In a future article I'll step through the calculation of variance, but for now, let's look at what differences in variance can do to your chance of winning.

We can arbitrarily divide video poker games into three categories: Low Variance (variance less than 25); Higher Variance (variance between 30 and 45), and Stomach-Turning Variance (variance greater than 50).

Low Variance games would include Jacks or Better, Bonus Poker, and Pick'em. Higher Variance games would include Double Double Bonus, Bonus Poker Deluxe, and White Hot Aces. Stomach-Turning Variance games would include Super Aces, Loose Deuces, and Triple Double Bonus. (Most Deuces Wild variations and Double Bonus games fall between what I'm calling Low Variance and Higher Variance.)

As representatives of each group, we can look at these three games: 9/6 Jacks or Better ("JOB"), 9/6 Double Double Bonus ("DDB"), and 8/5 Super Aces ("SA"). The variances of those three games are 19.5, 42.0, and 63.4.

How much money would we need to play each of those games at a 25c level for six hours at 500 hands/hour? (Assume that we're willing to accept no more than a 2% chance of going broke.) Is $400 enough? How about $600 or maybe $1000? The answer, of course, depends on which game we want to play.

Four hundred dollars is enough for six hours of JOB. What if you brought $400, but decide to play 9/6 DDB instead? In that case, you‘d have a 30% chance of busting out before the six hours were up. If you chose Super Aces, you'd have a 40% chance of busting out. You'd need $700 to play DDB for six hours. But even $700 is not enough for Super Aces. For SA you'd need $800. So you need twice as much bankroll to make it through six hours of Super Aces as you would playing JOB. Bottom line: When you sit down at a high variance game, you should assume you'll need substantially more bankroll.

Sometimes the decision about which game to play will depend on how much value you put on winning, versus how much importance you give to not going broke. For example, if you have $400 and try to play six hours of Super Aces, you'll have a 40% chance of going broke, but you'll have a reasonable 39% chance of ending up a winner. Compare that to 9/6 JOB, where you have a much smaller chance of busting out (just 1%), but where you also have a much smaller chance of ending a winner (29%). So, if your top priority is winning and you brought $400, you'd want to play Super Aces, despite the higher chance of going broke. If your top priority is getting to play for the full six hours and still having some money at the end, then you'd want to play JOB.

Most video poker players have an understanding of the importance of a game's expected value ("EV"). The importance of variance and the interplay between variance and EV on your outcomes is less well understood.

This interplay between EV and variance can be complex. For short plays, a game with worse EV may allow you to be a winner more often than a game with better EV but higher variance. For longer plays, the games with better EV will generally make you a winner more often. Sometimes the higher variance (and better EV) game will offer a greater chance of ending a winner, despite also carrying a large chance of losing everything.

Dunbar

(h_dunbar@hotmail.com)