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Ram has a fair coin, i.e., a toss of the coin results in either head or tail and each event happens with probability exactly half $(1/2)$. He repeatedly tosses the coin until he gets heads in two consecutive tosses. The expected number of coin tosses that Ram does is.

  1. $2$
  2. $4$
  3. $6$
  4. $8$
  5. None of the above
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5 Answers

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We need 2 consecutive head in 1 coin

out can be 

n=2             {HH}=1/22

n=3             {THH}=1/23

n=4             {HTHH,TTHH}=2/24

n=5              {TTTHH,THTHH,HTTHH}=3/25

....................................

 So, Expected no. of tosses (1/22+1/23+2/24+3/25+................)

                                             =1/22+ 1/22(1/2+2/22+3/23+.......................)

                                             =1/4 +1/4*2

                                             =1/4+1/2

                                             =3/4

Ans is (e)

http://math.stackexchange.com/questions/337937/why-sum-k-1-infty-frack2k-2

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