I'm familiar with the problem and my intuition is still wrong.
I wonder if you couldn't design a gambling machine that has a variant of the Monty Hall problem built-in favoring the house (naturally).
I wonder if you couldn't design a gambling machine that has a variant of the Monty Hall problem built-in favoring the house (naturally).
Imagine you're lost somewhere in the entire universe, trying to find your way home. Friendly (but tricksy) aliens offer you a deal. First, you pick some coordinates.
(Possibility 1) If the coordinates you pick aren't on earth, they'll take you to earth.
(Possibility 2) If the coordinates you pick happen to be on earth, they'll take you god knows where.
You also have the option to be taken to the coordinates you picked.
As a lost spaceman, perhaps going by the name Arthur Dent, what's your most likely path home? Odds are, if you pick a random point in the universe, it's not on earth. The odds are literally astronomical. So you're almost certainly going to find yourself in possibility 1.
Or just imagine monty hall with infinite doors. I'm thinking of a number between 1 and infinity (secretly, it's 19083412039102388171230123). You pick a number, and then I'll narrow down your choice to two options. Do you think you just happened to pick my number, or do you switch?
When you first pick a door, you have a 1/3 chance of it being the right door. There's a 2/3 chance of it being behind a door you didn't pick.
When the host then opens a door, there's still a 2/3 chance that it is behind one of the doors you didn't pick. However, there's now only one door in this set, so there's 2/3 chance that it's behind _that_ door.
To look at another way, imagine if the host didn't reveal the content of the door, but gave you the option to switch to BOTH of the other doors instead of your door. Your odds of winning clearly go up, as you now have two chances to win (and all are of equal probability). That's equivalent to what's happening here. By showing you the losing door of those two, he doesn't change anything - there's still twice the chance it was behind one of the doors you didn't pick compared to the one you did, and by switching you win if it was behind either of them.
The machines are random, with no enforcement on wins per time interval.
Source: I work in the industry.
Oh, and fwiw, I know several people who always win when they come to Las Vegas. I know because they've told me so. This proves that you don't actually work in the industry.
(I really hate that I have to disclose that I don't really have a Ph.D. is slot machines.)
If you like, add a bit of skill that gives the primates the feeling of control (video poker) or when you let loose of the handle or press a button (which simply grabs a number).
Stir in a bit of marketing pizzazz (off-by-one on slots, enormous number of simultaneous smaller bets, low payoff better odds vs. high payoff worse odds) and you can get the mark to stand there for a bit longer.
Networking, recognition and databases on players, etc. add more wrinkles.
In any case, I don't doubt that a Monty Hall bet could be wrappered into the game somehow to give the appearance of better odds than really exist.
(hopefully I got this somewhat right, I've never designed any gaming stuff. I'll bet that a gig at Bally or IGT would be a real eye-opener.)