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probability

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Suppose an unfair coin comes up heads 52.2% of the time if it is flipped. If the coin is flipped 26 times, what is the probability that:

a) it comes up tails exactly 12 times?
b) it comes up heads more than 22 times?

  • probability -

    This is a binomial expansion with p=0.522, and q=1-p=0.478.
    We will calculate the terms of
    (p+q)^26.
    Using the notation
    (n,r)=n!/((n-r)!r!)=n choose r
    the binomial expansion can be expressed as
    (p+q)^26
    =p^26+(26,1)p^25q+(26,2)p^24q^2+...+(26,r)p^(26-r)q^r...+(26,1)pq^25+(26,0)q^26

    P(12 tails)
    =P(14 heads)
    =(26,12)p^(12)q^(14)
    =0.1285...
    P(>22)
    =P(23)+P(24)+P(25)+P(26)
    =(26,3)p^23q^3+(26,2)p^24q^2+(26,1)p^25q+(26,0)p^26
    =0.00001759...

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