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 a/b, where a and b are coprime positive integers. What is the value of a+b?
To find the probability of getting exactly two heads, given that at least one flip results in a head, we can use conditional probability.
Let's consider the possible outcomes when flipping a fair coin 3 times:
HHH
HHT
HTH
HTT
THH
THT
TTH
TTT
Out of these 8 possible outcomes, we need to find the probability of getting exactly two heads, given that at least one flip results in a head.
First, let's find the probability of getting at least one head. This can be calculated by finding the probability of getting all tails and subtracting it from 1.
The probability of getting all tails = 1/2 * 1/2 * 1/2 = 1/8
So, the probability of getting at least one head = 1 - 1/8 = 7/8
Now, let's find the probability of getting exactly two heads and at least one flip resulting in a head.
Out of the possible outcomes, we can identify the favorable outcomes:
HHH
HHT
HTH
THH
Therefore, the probability of getting exactly two heads and at least one flip resulting in a head is 4/8 = 1/2.
Finally, the probability of getting exactly two heads, given that at least one flip results in a head, can be calculated using conditional probability:
P(Exactly 2 heads|At least 1 head) = P(Exactly 2 heads and at least 1 head) / P(At least 1 head)
P(Exactly 2 heads|At least 1 head) = (1/2) / (7/8)
= 8/14
= 4/7
So, the value of a is 4 and the value of b is 7.
The value of a+b = 4+7 = 11.
Therefore, the value of a+b is 11.