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Probability - expectation and variance from a coin toss

  1. Jan 30, 2014 #1
    1. The problem statement, all variables and given/known data

    A coin is flipped repeatedly with probability [tex]p[/tex] of landing on heads each flip.

    Calculate the average [tex]<n>[/tex] and the variance [tex]\sigma^2 = <n^2> - <n>^2[/tex] of the attempt n at which heads appears for the first time.


    2. Relevant equations

    [tex]\sigma^2 = <n^2> - <n>^2[/tex]

    3. The attempt at a solution
    I have the probability that head will appear for the first time on the nth attempt to be [tex]p(1-p)^{n-1}[/tex]. Aside from that I'm not sure where to go.
     
  2. jcsd
  3. Jan 30, 2014 #2

    jbunniii

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    Your expression for the probability that the head occurs for the first time on the ##n##th attempt looks correct to me. To find ##\langle n \rangle##, plug your expression into the definition of the expected value:
    $$\sum_{n=1}^{\infty} n p(1-p)^{n-1}$$
    To evaluate the sum, try to express it in terms of a power series.
     
  4. Jan 31, 2014 #3

    Ray Vickson

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    There are more-or-less standard formulas for sums like ##\sum_{n=1}^{\infty} n x^n## and
    ##\sum_{n=1}^{\infty} n^2 x^n##. These may be found in books, and in on-line sources. Basically, they are easy to get manually, by looking at ##S(x) = \sum_{n=1}^{\infty} x^n## and then looking at what you get from ##dS/dx##, etc.
     
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