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Ask the Wizard #234

I have been in the casino business for 27 years and have worked positions from dealer to casino manager. I am dealing now, and there is a player that comes in our casino and plays blackjack who has won $5,000 or more, for at least 30 consecutive trips. Having been a manager, I know that even the best card counter does not win every time. I believe it is not possible to win that many trips in a row and not be cheating somehow. I have only dealt to him personally one time, and I was unable to detect anything out of the ordinary. He has a $35,000 credit line and sometimes starts off losing, but he always winds up winning in the end. He bets anywhere from $200 to $5,000 and doesn’t appear to be counting cards. Do you think it is possible to win so many times in a row without cheating?

Daniel from Philadelphia

Let's ignore the house edge for the sake of simplicity. If the player’s two stopping markers are a win of $5,000 or a loss of $35,000, then the probability of reaching the winning marker is 7/8. The probability of doing that during 30 consecutive trips is (7/8)30 = 1.82%. So, this could easily be just good luck. I’d let him play until you determine why and how he is beating you.

This brings to mind a very good book I just read, Casino-ology, by Bill Zender. The main thrust of the book is that casino management is much too paranoid about advantage play. The overreaction to it is slowing down play and annoying legitimate customers, the cost of which is much more than what is saved by catching a few extra advantage players.

Suppose you are playing standard 80-spot Keno with 20 drops, but the drops are "with replacement." That is, after a ball is dropped, its number is recorded and it is put back in the hopper and may be drawn again. Suppose you mark a card with 4 spots. What are the probabilities of 0, 1, 2, 3 and 4 distinct hits?

Eliot from Santa Barbara

That is actually a pretty hard problem. It is easy to get the probability of the number of times any of your four balls are drawn, including repeats. The tricky part is determining the probability that x distinct picks will be chosen, given that any pick was chosen y times. I indicate the answer and solution on my MathProblems.info page, problem 205.

I’ve had good luck betting with teams that have had a streak of 5 losses and against teams that have won 5 in a row. I argue that unlike dice and roulette, where you say that the dice or ball has no memory, a professional sports team cannot win or lose endlessly. I chose 5 in a row after analyzing when the switch would happen more often. It doesn’t matter to me where they are playing or who is the starting pitcher, injuries, etc. Do you think I am correct in my thoughts?

Matt

I think these are good bets, but not for the reason you state. A sports team can, in theory, lose endlessly. One reason these are good bets anyway is they are going to be on big underdogs the vast majority of the time, and underdogs are generally stronger bets than favorites. For another reason, the square bettors are generally betting the other way in these situations, creating value your way.

My question is about the history of California gaming law, especially craps and roulette. Do you know why California does not allow the dice or the ball and wheel to directly determine the outcome of their respective games? Or do you know where this law originated? What were California law makers thinking when they wrote this?

Chrs from Chula Vista

Sorry, I don’t know the history or reason behind that law. It was probably a misguided compromise between puritan and gambling interests. They were likely thinking the same kind of thing Mississippi lawmakers were when they only permitted non-Indian gaming on “riverboats.” We all saw the result of that brilliant idea after Hurricane Katrina. As I’ve been saying for years, my opinion is if you’re going to allow gambling, then drop the pretenses and allow it the whole way.

Given that the odds of a Royal in full pay deuces wild is approximately 1 in 49,000, what is the probability of hitting exactly two in 6,000 hands? My wife and I just returned from the Red Rock, where I did this.

Don from Raleigh

The probability of hitting a royal is actually 1 in 45,282 per hand. The probability of hitting exactly two royals in 6,000 hands is combin(6000,2)×(1/45282)2×(45281/45282)5998 = 0.007688177, or 1 in 130.