You have 100 dollars, and there is a dollar bill behind each door. You roll a 100 sided die 100 times, and you take the dollar behind the door on the die roll if the bill has not been taken already (e. g. you roll 16, then you take the dollar behind door 16 if you haven't already taken it). What is your expected payoff?

63.21.

Step-by-step explanation:

You have 100 dollars, and there is a dollar bill behind each door. You roll a 100 sided die 100 times, and you take the dollar behind the door on the die roll if the bill has not been taken already (e. g. you roll 16, then you take the dollar behind door 16 if you haven't already taken it). What is your expected payoff?

X=Σ100i=1 1Ai

where Ai is the event that door i is opened at least once, and 1Ai is the indicator function for event Ai.

Thus the expected payoff is:

E[X]=Σ100 as i=1 Pr[Ai].

to calculate Pr[Ai].

Ai∁ is the probability of the event that after 100 rolls, door i is not chosen, which is:

Pr[Ai∁] = (99/100) ^100

Thus:

Pr[Ai]=1-Pr[Ai∁]=1 - (99/100) ^100.

E[X]=Σ100i=1Pr[Ai]=100 * (1 - (99/100) ^100).

Also based on the following approximation for large n's:

(1-1n) n≈1e

we have:

(99/100) ^100 = (1-1/100) ^100≈1/e.

The expected pay off is

E[X]=100 * (1 - (99/100) ^100) ≈100 * (1-1/e) ≈63.21.