4 Rolls To 4 [Visa Statistics Interview Question]

A fair die is rolled repeatedly until a number larger than 4 is observed.

What's the probability that it takes you 4 rolls to roll a number greater than 4?

Taking 4 rolls to roll a number greater than 4 means that on the first 3 rolls you rolled 4 or less, and the last roll you rolled a number greater than 4.

The probability of rolling 4 or less: $2/3$

The probability of rolling 4 or less, 3 times in a row, is: $(2/3)^{3}$

The probability of rolling 4 or less 3 times in a row, combined with rolling above a 4 on the 4th roll is:

$(2/3)^{3} * (1/3) = 8/81$