Here's a neat one, apologies if it's been seen before.
I suggest we play a game. I am using a fair coin.
On round 1, I toss the coin, and if it's heads, I give you 1 euro. If it's tails, we move to round 2.
On round 2, the same thing happens, but if it's heads I give you 2 euros. Again, if it's tails we move to round 3.
On round 3, I'll give you 4 euros if it's heads.
On round 4, 8 euros, etc. The payoff for heads at round n is 2^(n-1) euros.
This continues until the coin comes up heads.
Question 1: What's the expected payoff for this game?
Question 2: How much would you _pay me_ for the opportunity to play the game?
Thursday, April 18, 2002
If Pn is the probabilily that the game ends on round n, then the chance I win is P2 + P4 + P6 + P8 + .. P2n ...
P2 = .5 * .5
P3 = .5 * .5 * .5
P4 = .5 * .5 * .5 * .5
Pn = 0.5^(n)
if En is the amount I expect to win by finishing at round n, then
E1 = 1
E2 = 2
E3 = 4
E4 = 8
En = 2^(n-1)
EnPn = (2^(n-1))* (.5^n) = 1/2
My expected winnings are the sum of EnPns that I win minus the ones that you win
My expected winnings are therefore -1/2 + 1/2 - 1/2 + 1/2 ... , which doesn't actually converge, but hovers between 0 and -1/2.
I wouldn't pay you a penny, or a eurocent if it comes to that.
Friday, April 19, 2002
Oops. I misread the question, thinking I only won on goes 2,4,6 etc. If I can win on any go, then my expected payoff is 1/2 + 1/2 + 1/2 + ... which is infinite.
Theoretically then, no amount would be to much to pay you. In practice, however, I probably wouldn't be willing to give you more than about 10 Euros, because I don't like losing money. :-)
Saturday, April 20, 2002
> How much would you pay me for the opportunity to play the game?
To be fair, I should pay you log2 of your wealth. Even though the series is mathematically infinite, you'll clearly default if it gets far enough along to exceed your wealth.
This by the way is why casinos love people who double their bets when they lose. The mark thinks that by doing this he's guaranteed a $1 winning. But the casino knows that these frequent $1 winnings will be more than offset by the occasional guy who loses his shirt.
Monday, April 22, 2002
Actually, casinos have a maximum limit to prevent the doubling trick, and/or forbid betting on credit. If I could bet on credit and had no limit, I'd have no fear of betting 2^100 or more dollars just to recoup my losses.
Tuesday, April 23, 2002
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