Question

. (Pitman 3.4.9) Suppose we play the following game based on tosses of a fair coin. You pay me $10, and I agree to pay you $n 2 if heads comes up first on the nth toss. If we play this game repeatedly, how much money do you expect to win or lose per game over the long run

Answer 1

Answer:

$6 per game

Explanation:

The probability of getting a head on a toss is given as 0.5 for a fair coin.

Therefore the expected number of times that the coin would be tossed to get the first head would be given as the expected value of the geometric distribution with parameter of p = 0.5. therefore the expected value here would be 1/0.5 = 2

Therefore, we expect to get 22 = 4 dollars but we paid initially $10, therefore in long run we expect to lose $6 per game.

Answer 2

Question

Suppose we play the following game based on tosses of a fair coin. You pay me $10, and I agree to pay you $n² if heads comes up first on the nth toss. If we play this game repeatedly, how much money do you expect to win or lose per game over the long run

Answer:

Lose of $4 per game

Explanation:

Let X represents the toss on which the first head appears.

Then X has a geometric distribution with parameter

p = ½

q = 1 - p

q = 1 - ½

q = ½

and you expect to be paid

A =∞ to n=1 for n² * P(X=n) =E(X²)

Where E(X²) = (1 + q)/p²

E(X²) = (1 + ½)/½²

= 3/2 ÷ ¼

= 3/2 * 4/1

= 12/2

= 6

Since you pay $10 to play the game,

you expect to lose $10 - $6 per game.

= $4