Why can this question be treated as a question with geometric random variable without any modifications?In an infinite sequence of flipping a fair coin - 0.5 probability to get heads/tails. (every flip is independent from the others).What is the expected value of number of flips until we get Heads then Tails?

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Zoey Benitez · Accepted Answer

Step 1Let       ξ    i   be the result of the i-th coin toss, and let   X  =      min          n        {  n  &amp;gt;  0  :      ξ    n    =  T  ,      ξ          n      −      1        −  H  }. You can use the law of total probability, then you have      E    (  X  )  =      E    (  X      |        ξ    1    =  H  )      P    (      ξ    1    =  H  )  +      E    (  X      |        ξ    1    =  T  )      P    (      ξ    1    =  T  )Step 2We can calculate the conditional expectations.  X      |        ξ    1    =  H  ∼  1  +  G  e  o  (  1      /    2  )    ⟹        E    (  X      |        ξ    1    =  H  )  =  1  +  2  =  3  E  (  X      |        ξ    1    =  T  )  =  1  +  E  (  X  )If we substitute the results in the above equation, we get:      E    (  X  )  =  3      1    2    +      (    E  (  X  )  +  1  )      1    2        E    (  X  )  =  4

szklanovqq · Accepted Answer

Step 1Say X is the event of getting H,T in a row. Based on the first flip, we have two casesi) The first flip is H - Then we are seeking the next flip to be a T. If we get T, we are done but if we get H, we are again looking for a T and so on.ii) The first flip is T - We spent a flip and we are again back to where we started, that is seeking H,T.Step 2So,   E  (  X  )  =  1  +      1    2    E  (  X  )  +      1    2    E  (  T  )Now we know   E  (  T  )  =  2So,   E  (  X  )  =  4

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