Question

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?

Answer 1

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...