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Math Help - Probability Coin Question

  1. #1
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    Probability Coin Question

    I flip a fair coin. What is the expected number of attempts it will take for one heads to show up and one tails to show up?

    Attempt at a solution: I have been trying to use conditional probability, with E(X) = (sigma from k = 1 to infinity)(E(X|Y = k))(P(Y = k))

    Any help is appreciated.
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  2. #2
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    Quote Originally Posted by noles2188 View Post
    I flip a fair coin. What is the expected number of attempts it will take for one heads to show up and one tails to show up?

    Attempt at a solution: I have been trying to use conditional probability, with E(X) = (sigma from k = 1 to infinity)(E(X|Y = k))(P(Y = k))

    Any help is appreciated.
    I guess you can read up on coupon collector problem - that shd give you enough hint. Also I'm not clear about your question, what do you mean by - "one heads to show up and one tails to show up". Plz define the event more precisely.
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  3. #3
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    how many flips will it take for at least one head to appear and at least one tail to appear?
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  4. #4
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    Quote Originally Posted by noles2188 View Post
    how many flips will it take for at least one head to appear and at least one tail to appear?
    Ok. So please read up
    Coupon collector's problem - Wikipedia, the free encyclopedia
    Answer to your question is a direct application of that formula.

    Another way to look at it is
    There are to mutually exclusive and exhaustive events
    E1: H....... (i.e. first flip is H)
    E2: T....... (i.e first flip is T)
    p(E1)=p(E2)=0.5

    Consider only E1. After the 1st flip you need a T. What is the expected number of flips to get a T? Let this be n. (What is n?)
    So expected number of flips to get atleast H and T = n+1

    Similarly for E2. (use symmetry)

    Final answer = p(E1)(n+1) + p(E2)(n+1) = n+1.
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