Posted by **Abhishek Ranjan** on Monday, June 10, 2013 at 1:35am.

A fair coin is flipped 3 times. The probability of getting exactly two heads, given that at least one flip results in a head, can be written as ab, where a and b are coprime positive integers. What is the value of a+b?

- Geometry -
**MathMate**, Monday, June 10, 2013 at 5:21pm
It is a conditional probability problem where the distribution is binomial.

The general expression for conditional probability for probability of event A given event B is

P(A|B)=P(A∩B)/P(B)

where A = P(exactly 2 heads)

B=P(at least one head)

P(A∩B)=P(A) since A is a subset of B.

Assuming a fair coin, p=1/2, q=1-1/2=1/2

P(A∩B)

=P(A)=P(exactly 2 heads)

=3C2 p^2 q^1

=3*(1/2)^2 (1/2)

=3/8

P(B)=P(at least one heads)

=1-P(no heads)

=1- 3C0 (1/2)^0 (1/2)^3

=1- 1/8

=7/8

So

P(A|B)

=P(exactly two heads | at least one heads)

=(3/8) / (7/8)

=3/7

- Geometry -
**Abhishek Ranjan**, Thursday, June 13, 2013 at 1:45am
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