A Gambling Question.
Let's say a casino (could it be one in St. Petersburg?) offers you the chance to play the following game in exchange for an ante. The game is very simple. The casino will flip a coin repeatedly until it comes up heads. You will get paid 2^n dollars, where n is the number of tails that came up consecutively before it comes up heads. For example, if the flips are TTTH, you get eight dollars. If it is TTTTH, you get sixteen. If the first flip is a head, you get one dollar (two to the zero is one), and so on.
The question is this: what is the maximum ante you would be willing to pay to play this game? Of course, it should be at least a dollar, since you are guaranteed to WIN at least a dollar, with a chance of winning more. But how much?
If you have an answer to that, ask yourself this: if you were the casino, what is the LEAST amount you would accept to play this game?
3 Comments:
I don't have the statistical tools or prowess at my disposal, but I'm sure you do:
The idea behind any casino game is to lure customers in with the promise of a possible big payoff, and enough small payoffs to keep them playing. The house simply wants a modest edge over time, so as people keep playing it keeps making money.
It seems like there should be some way of figuring out what the expected payout over time would be (50 percent of the time the player would get $1, then my math skills are fuzzy. Is it 75 percent of the time the player would get either $1 or $2? I don't know how to calculate what the average payout would be, or whatever the most relevant measurement is. Some sort of limit calculation, I suppose). So I don't know what I'd charge to have people play the game.
But based on no math calculations whatsoever, I would say I'd put down $5 with no hesitation. I'd probably be willing to play if the ante was more, after a few drinks, but I think $10 would be the absolute limit.
If I'm doing my math right, then here are my answers:
if I'm the "house" -- I'd say, I'll end the game at 10 tosses, to limit my losses. for 1000 people, I'd expect to pay out about $5000, so I want people to ante 6.
if I'm playing, then I see that I'd have a 1:8 chance of winning back my $6 plus $2 in winnings.
(I won't play any game where the house has a steady advantage, and no amount of skill can make up for it)
But, it is interesting to see how the increment per-round raises the winnings by an amount in step with the drop in likelihood of winning.
I don't think any house would want any risk of being on the line for such huge winnings, however.
The expected value of your winnings in this game is infinite, right?
In which case, especially if I'm in St. Petersburg, there's no way I'm going to play. The existence of the casino is a strong indication to me that I am not being given a chance to play the game I think I'm playing, or that it is quickly followed by quick round of another game that involves great risk to life and limb.
JRS
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