A fair six-sided die (with faces numbered 1, 2, 3, 4, 5, 6) is repeatedly thrown independently. What is the expected num...

Question

A fair six-sided die (with faces numbered 1, 2, 3, 4, 5, 6) is repeatedly thrown independently. What is the expected number of times the die is thrown until two consecutive throws of even numbers are seen?

A. 8
B. 4
C. 6
D. 8

Accepted Answer

The expected number of throws is 6. We roll a fair die repeatedly and stop when we see two consecutive even numbers. Even faces are 2, 4, 6 so P(even) = 1/2. We need the expected number of rolls to get two even numbers in a row. This is a classic Markov chain / conditional expectation problem with states tracking whether the last roll was even.

A fair six-sided die (with faces numbered 1, 2, 3, 4, 5, 6) is repeatedly thrown independently. What is the expected number of times the die is thrown until two consecutive throws of even numbers are seen? A. 8 B. 4 C. 6 D. 8