Question

You roll a six-sided die repeatedly until you roll a one. Let X be the random number of times you roll the dice. Find the following expectation:

E[(1/2)^X]

Answer 1

X is the number of times a six sided die is rolled until we get a one. Therefore the distribution for X here is obtained as:

P(X = 1) = 1/6
P(X = 2) = (5/6)*(1/6)
P(X = x) = (5/6)^x-1*(1/6)

The expected value of (1/2)^x is computed here as:

Applying the sum of an infinite geometric progression, we get here:

Therefore 0.1429 is the required expected value here.