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GATE DA 2024 Q36: Expected throws until two consecutive even numbers

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?

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Answer: C (official GATE 2024 final answer key)

Concept used: Expected waiting time by first-step analysis (states: no even yet, last throw even).

First step: With p=12p = \tfrac{1}{2}, let E0E_0 and E1E_1 be the expected remaining throws; E0=1+12E1+12E0E_0 = 1 + \tfrac{1}{2}E_1 + \tfrac{1}{2}E_0 and E1=1+12E0E_1 = 1 + \tfrac{1}{2}E_0.

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Source: official GATE 2024 DA question paper and final answer key, organised by IISc Bangalore. Question paper · Answer key

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