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?

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

10

To find the probability of getting exactly two heads, given that at least one flip results in a head, we can use conditional probability.

First, let's find the probability of getting at least one head. The probability of getting at least one head is equal to 1 minus the probability of getting all tails.

The probability of getting all tails in three coin flips is (1/2) * (1/2) * (1/2) = 1/8.

Therefore, the probability of getting at least one head is 1 - 1/8 = 7/8.

Now, let's find the probability of getting exactly two heads, given that at least one flip results in a head.

To do this, we need to consider the cases where we get two heads and where we get three heads, as both of these cases satisfy the condition of at least one flip resulting in a head.

Case 1: Two heads and one tail
The probability of getting exactly two heads and one tail is given by (3 choose 2) * (1/2)^2 * (1/2) = 3/8. (3 choose 2) represents the number of ways we can choose 2 out of 3 flips to be heads.

Case 2: Three heads
The probability of getting exactly three heads is (1/2)^3 = 1/8.

Therefore, the total probability of getting exactly two heads, given that at least one flip results in a head, is (3/8) + (1/8) = 4/8 = 1/2.

The probability can be written as ab, where a = 1 and b = 2. Therefore, the value of a + b = 1 + 2 = 3.