Question
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?
Answers
MathMate
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
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
Abhishek Ranjan
10