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Ask the Wizard #199

Bally Gaming has a single-deck, multi-hand, blackjack game. The player plays seven hands against a single dealer hand. There is an interesting rule in that if the game runs out of cards, all unbusted player hands automatically win. What is the probability of running out of cards? Can have suggest any strategy changes to run out the deck?

Michael L. from West Mifflin, PA

For the benefit of other readers, the full set of rules is:

  • Single deck.
  • Dealer stands on soft 17.
  • Winning blackjack pays even money.
  • Player may double any first two cards.
  • No double after split.
  • Player may resplit to four hands, including aces.
  • No draw to split aces.
  • No surrender.
  • Six-card Charlie (player unbusted six cards automatically wins).
  • Cards shuffled after every hand.
  • If game runs out of cards, all unbusted player hands automatically win.

The house edge using total-dependent basic strategy is 2.13%. I ran a 7-player simulation, using total-dependent basic strategy, and the average number of cards used per round was 21.65, with a standard deviation of 2.72. In almost 190 million rounds played, the most cards ever used was 42, which happened 7 times.

It is my educated opinion that even with computer perfect composition-dependent strategy the player would still realistically never see the last card. You could cut down the house edge much more using composition-dependent strategy, according to all the cards seen as you go along. However bucking 2.13% house edge to start with, you’ll never get anywhere near break-even, regardless of how hard you try.

Is there a statistical test to check that a slot machine’s payout is correct? For example, the casino claims 93% payout, but a test shows 91% payout in 10,000 games. I think statistically, this may be okay, but I don’t know how the math would work.

Mary Jo from Calgary

Let’s assume 10.8 for the standard deviation, which I get from the Red, White, and Blue game described in my slot machine page. The standard deviation of the mean over n spins is standard deviation per bet divided by the square root of n. In this case, 10.8/10,0000.5 = 0.108. The difference between 93% and 91% over 10,000 spins is just 18.5% of one standard deviation. To get the standard deviation of the mean to just 2% you would need a sample size of 291,600 spins. The standard deviation in slots will vary substantially, so take these figures with a grain of salt.

Could you say a few words about the controls and consumer protection in cruise ship casinos of major lines. Are there avenues to make a protest or review?

Steve S. from Lake Grove, NY

To be honest with you, I don’t know that much about it. I would imagine you would have to file a complaint through the country in which the ship is registered, usually Panama, the Bahamas, or Liberia. Good luck getting any satisfaction that way. Your odds would probably be better writing to the corporate headquarters of the cruise line. As a last resort, I would suggest making a stink on as many forums about cruising as you can.

Sir, thank you so much for such a wonderfully informative site. Could you comment on the variance and covariance in Spin Poker

J.B. from Las Vegas

You’re welcome. I ran some random simulations in 9/6 Jacks or Better, to get at the answer to your question. The following table shows the covariance for 2 to 9 lines played, in 9/6 Jacks or Better. The variance would be the same as the base game played.

Covariance in 9/6 Jacks or Better Spin Poker

Lines Covariance
2 1.99
3 3.70
4 9.62
5 15.27
6 19.53
7 23.37
8 27.94
9 33.46

Let’s look at an example of 9-line 9/6 Jacks or Better. The variance of the base game is 19.52. The covariance is 33.46. So the total variance is 19.52 + 33.46 = 52.98. The standard deviation is 52.981/2 = 7.28.

Why does BetFair have zero house edge blackjack? I suspect that it is because they realize that players either don’t know the optimal strategy or sometimes can’t act on it (for example if they bet their last chip and then are unable to act on a double down or splitting opportunity).

Nick from London

They also have zero house edge baccarat and roulette, so that can’t be entirely the reason. My theory is that it is a way to get players through the door. Their main casino has many more games and higher bet limits. I’m sure some of the Zero Lounge players wander into the regular casino eventually.

I have looked at many video poker strategy charts, and many are different. Are they, or should they be, the same based only on probabilities and nothing else? I asked one author and he said that he "tweaked" the charts, but gave no method.

Jack from Georgetown

Video Poker strategy charts are not an exact science. There is always a tradeoff between brevity and accuracy. There are also issues about the best way to express a rule. Unless there was a huge emphasis on simplicity, it is unlikely two writers would come up with the same strategy.

I visited Bodog and tried out their roulette wheel on the free site. In a box in the upper corner it records the last ten numbers that hit. I spun the thing fewer than 20 times, I am sure. The numbers recorded there are as follows: 9-9-29-21-11-11-20-28-32-1 Interestingly, two spins before this there was another hit on 32. Meaning that the numbers 9, 11 & 32 all hit twice within 12 spins. As I said, I am not a statistician, but the frequency of these three numbers coupled with the minimal number of times I spun the wheel seem to indicate something is wrong.


The probability of three pairs and six singletons in twelve spins is combin(38,3) × combin(35,6) × combin(12,2) × combin(10,2) × combin(8,2) × fact(6)/3812 = 9.04%. The math gets rather messy asking about the probability that this could happen in any 12-spin span over 20 total spins. Suffice it to say that it is significantly more than 9%, more likely than not, I would guess. So these seem like very normal results to me.