GATE EC 2015 Set 2 — Question 49
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Engineering Mathematics → Probability → Expectation & Variance
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Question
Let the random variable represent the number of times a fair coin needs to be tossed till two consecutive heads appear for the first time. The expectation of is _______.
Correct answer
6 to 6
Solution
Let be the expected number of tosses to get two consecutive heads (HH).
Let be the expected number of additional tosses needed when we have 0 consecutive heads.
Let be the expected number of additional tosses needed when we have 1 consecutive head.
We want to find .From :
Let be the expected number of additional tosses needed when we have 0 consecutive heads.
Let be the expected number of additional tosses needed when we have 1 consecutive head.
We want to find .From :
- If we toss T (prob 1/2), we stay at 0 consecutive heads. Tosses used: 1.
- If we toss H (prob 1/2), we move to 1 consecutive head. Tosses used: 1.
- If we toss T (prob 1/2), we go back to 0 consecutive heads. Tosses used: 1.
- If we toss H (prob 1/2), we reach the goal (HH). Tosses used: 1.
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