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Expected Value: Coin flip not same as previous flip

  1. Mar 15, 2017 #1
    1. The problem statement, all variables and given/known data
    Given a sequence of Heads and Tails, let’s say that the sequence has a switch each time one toss is different than the toss before it. For instance, the sequence HHTHTTTHTH has 6 switches.

    Suppose you toss a fair coin N times and record the resulting sequence of Heads and Tails. If X is the number of switches in the sequence, find E(X).

    2. Relevant equations
    We are assuming that the expected value, E(X) = ∑x_i*P(X=x_i) (the sum of each element of X, multiplied by the respective probability of that element occurring)

    3. The attempt at a solution
    I understand that this is going to have something to do with the number of heads and tails in the sequence and the relationship between the two. Clearly 2/n of the sequences have 0 switches, and 2/n of the sequences have n-1 switches, however I'm unsure of how to work out the values in between in general.
     
  2. jcsd
  3. Mar 16, 2017 #2
    it's rather simple. use E(X+Y)=E(X)+E(Y)
     
  4. Mar 16, 2017 #3

    PeroK

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    Can you see how to do it without doing any calculations? Hint: what is the expected number of Heads in a sequence of tosses?
     
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