Posted by **Sam** on Wednesday, March 21, 2012 at 7:55pm.

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 -
**MathMate**, Wednesday, March 21, 2012 at 9:49pm
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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