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 break down the problem step by step:
Step 1: Determine the sample space
When a fair coin is flipped 3 times, each flip has two possible outcomes: heads (H) or tails (T). So the sample space, S, consists of all possible sequences of three flips: {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}.
Step 2: Determine the event of interest
We are interested in the event that exactly two heads are obtained, given that at least one flip results in a head. Let's call this event A. The possible outcomes for event A are {HHH, HHT, HTH, THH}.
Step 3: Calculate the probability
To calculate the probability of event A occurring, given that at least one flip results in a head, we need to divide the number of favorable outcomes by the number of possible outcomes.
The number of favorable outcomes is 4, as there are four sequences in event A.
The number of possible outcomes is 7, as there are seven sequences that have at least one head: {HHH, HHT, HTH, HTT, THH, THT, TTH}.
Therefore, the probability of getting exactly two heads, given that at least one flip results in a head, is 4/7.
Since the numerator and denominator of the probability fraction (a and b) are already coprime positive integers, a=4 and b=7.
Finally, the value of a+b is 4+7=11.
So, the value of a+b is 11.