GATE DA 2024 Set 1 — Question 36
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Probability & Statistics → Discrete Random Variables → Expectation (Discrete)
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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?
Correct answer
(C) 6
Solution
Let be the expected number of throws to get two consecutive even numbers.
Let be the probability of getting an even number. Since the die is fair and faces are {1, 2, 3, 4, 5, 6}, the even numbers are {2, 4, 6}.
.
Let be the probability of getting an odd number.Let be the expected additional throws needed starting from state 0 (no consecutive evens so far).
Let be the expected additional throws needed starting from state 1 (one even number just thrown).From state 0:
(Equation 1)From state 1:
(Equation 2)Substitute (2) into (1):
The expected number of throws is 6.
Let be the probability of getting an even number. Since the die is fair and faces are {1, 2, 3, 4, 5, 6}, the even numbers are {2, 4, 6}.
.
Let be the probability of getting an odd number.Let be the expected additional throws needed starting from state 0 (no consecutive evens so far).
Let be the expected additional throws needed starting from state 1 (one even number just thrown).From state 0:
- If we throw Odd (prob ), we return to state 0. Cost: 1 throw + .
- If we throw Even (prob ), we move to state 1. Cost: 1 throw + .
(Equation 1)From state 1:
- If we throw Odd (prob ), the streak breaks, return to state 0. Cost: 1 throw + .
- If we throw Even (prob ), we reach the goal (2 consecutive evens). Cost: 1 throw + 0.
(Equation 2)Substitute (2) into (1):
The expected number of throws is 6.
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