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