Ask the Wizard #234
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.
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 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.
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.
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.