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Ask the Wizard #224

What is the coefficient of skewness for video poker?

David from Fort Worth, Texas

For the benefit of other readers, the coefficient of skewness (skew) for any random variable is a measure of which direction has the longer tail. A negative skew means the most likely outcomes are on the high side of the distribution, offset by the extremes tending to be on the low side. A positive skew is the opposite, where the most likely outcomes are the low side, but with the extremes tending to be on the high side. The mean is less than the median with a negative skew, and greater with a positive skew. An exact formula can be found at Wikipedia, or lots of statistics books.

Loosely stated, skewness is going to correlate with how often you get a win in a session. In Jacks or Better, for the most part, you are not going to get a winning session over a few hours if you do not hit a royal. You can sit down at Double Double Bonus and be a winner after a few hours more often because of the big quad payouts. Because most people are subject to cognitive biases, the pain from a loss is twice the pleasure from a win. People do not really play Double Double Bonus because they like the variance, they play because they have a greater shot at winning. The following table shows some key statistics for four common video poker games. It is interesting to note that skew is greatest for Jacks or Better.

Key Video Poker Statistics

Statistic JoB — 9/6 BP — 8/5 DDB — 9/6 DW — NSUD
Return 0.995439 0.99166 0.989808 0.997283
Variance 19.514676 20.904113 41.985037 25.780267
Skew 147.114643 134.412152 66.495372 101.23991
(Excess) Kurtosis 26,498 23,202 6,679 14,550

JoB — 9/6 = Full pay Jacks or Better
BP — 8/5 = Standard pay Bonus Poker
DDB — 9/6 = Standard pay Double Double Bonus Poker
DW — NSUD = "Not so Ugly Ducks" Deuces Wild

How can knowing this actually help the video poker player? I suppose one could say that a game with a large skew has a greater chance of a loss over a session of a few hours. For example, in Jacks or Better, if you don’t hit any royals, the house edge will probably eventually grind your bankroll down. However in a game like Deuces Wild or Double Double Bonus, the second highest wins can pull you out of the hole over a session. In other words, the skew keeps you from winning when you are not hitting royals. Knowing the skew won’t increase your odds, but it is mentally helpful to know what to expect. So, the next time you take a beating in 9/6 Jacks, blame it on the skew.

My thanks to Jeff B. for his help with this question.

What is the largest casino in the world?

James from Worchester, MA

Measured in terms of square feet of casino space, it is the Venetian in Macau. The most table games can be found at the Sands in Macau, and the most slots at Foxwoods in Connecticut. The following table shows some details.

Largest Casinos in the World

Property Locale Casino Feet2 Table Games Slots
Sands Macau 229,000 1000 750
Venetian Macau 456,000 870 3,000
Foxwoods Connecticut 344,000 380 7,200
Mohegan Sun Connecticut 300,000 392 6,780
MGM Las Vegas 170,000 178 2,523
Wynn Las Vegas 111,000 185 2,000


While playing pai gow poker, I was dealt a bad pai gow hand. As it turned out, the dealer had the identical numeric sets of cards, so I obviously lost $5. The dealer blurted out she had never seen that before and the pit girl agreed. What are the odds that the odds of a double copy, when the player has a pai gow?

Charles N. from Las Vegas

For the benefit of other readers, a "pai gow" hand is one with seven singletons, and no straight or flush is possible. The probability that the dealer will have the same seven ranks, without using the joker, is 37/combin(46,7) = 2,187/53,524,680 = 1 in 24,474.

How much do casino dealers make?

Larry from Las Vegas

The wages are pretty low, not much more than the minimum wage. The minimum wage in Nevada, as of this writing, it is $5.85/hour if health insurance is provided and $6.85 otherwise. However, tips can add to that significantly. At most casinos in Vegas, the dealers pool tips. The only exceptions I have heard of, where dealers keep their own tips, are Caesars Palace and Hooters. Following is a chart of average tips over an 8-hour shift for one week in November, 2008. These figures are quoted, with permission, from The Dealer's News. My thanks to Ron Saccavino.

Average Tips — November 2008

Property Avg. Tips
Wynn $263
Hard Rock $203
MGM $158
Green Valley Ranch $156
Harrah's $129
Sunset Station $125
Bill's $123
Planet Hollywood $118
South Point $113
Monte Carlo $111
Sahara $66
Stratosphere $65

Update: The Dealer's News appears to have been discontinued since this question was published.

I’m a pretty good poker player, and I’m interested in learning blackjack as well. The problem is, I’ve always been a bit slow, in the literal, non-euphemistic sense. I can’t count the cards at the speed they come out. I’ve tried, but I always wind up dropping the sign on negative numbers, or mentally count in the wrong direction.

I would have much better luck if I could just count the tens and never deal with negative numbers. That should be possible, right? The major factor is how "rich in tens" the deck is, right? I know you have your Ace-Five count on your site, but you say that’s profitable only if the dealer stands on soft 17, and that’s not the case at the casino I go to.

So my questions are: (a) Am I barking up the wrong tree entirely by thinking you could use just a count of tens (adjusted for number of decks left, of course)?, and (b) Do you know of any such strategy that doesn’t use negative numbers?

Peter from Redmond, WA

Lest any experts in blackjack history correct me, Lawrence Revere has a chapter on the Revere Ten Count Strategy in his classic work Playing Blackjack as a Business. This strategy requires keeping two counts, one of tens, and one of everything else. Play and bet amount is then dictated by the ratio of these two counts. In my opinion, this strategy is too difficult to use. I have yet to meet anybody who actually practices this count.

The Speed Count requires only counting low cards, offset by adjustments in the number of hands played. There are no adjustments for the number of decks left, known as a "true count conversion." If you really hate negative numbers, you will probably never see one with the Speed Count, although one subtraction per round is required. The effectiveness and ease of use are a topic of hot debate in the blackjack community. For more information, please read my review .

If negative numbers are the only issue, my webmaster Bluejay notes that you could adjust any counting system that does not involve a true count conversion, like the Knock Out, by starting at a higher number. Of course, you should adjust all the index numbers by the same amount.