The GATE Grind

GATE 2024 DA – Question 36

Probability and Statistics · Conditional expectation and variance · 2 marks · Multiple choice

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?

  1. 2
  2. 4
  3. 6
  4. 8

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Correct answer: (C) 6

Explanation

An even number has probability $p = \frac{1}{2}$, so this is like waiting for two heads in a row with a fair coin. Let $E$ be the expected number of throws from the start and $E_1$ the expected number after one even throw. $E = 1 + \frac{1}{2}E_1 + \frac{1}{2}E$ and $E_1 = 1 + \frac{1}{2}\cdot 0 + \frac{1}{2}E$. Substituting, $E = 1 + \frac{1}{2}\left(1 + \frac{E}{2}\right) + \frac{E}{2} = \frac{3}{2} + \frac{3E}{4}$, so $\frac{E}{4} = \frac{3}{2}$ and $E = 6$.