What is the probability of obtaining exactly two heads in three flips of a coin, given that at least one is a head?
To find the probability of obtaining exactly two heads in three flips of a coin, given that at least one is a head, we need to calculate the conditional probability.
Let's break down the problem step by step:
Step 1: Determine the total number of possible outcomes. In three flips of a coin, each flip can result in two possibilities, either heads or tails. So, the total number of outcomes is 2^3 = 8.
Step 2: Find the number of outcomes where at least one is a head. There are two possible scenarios: Either the first flip is a head, or the first flip is a tail and the second or third flips are heads. Let's calculate the number of outcomes for each scenario:
- Scenario 1: The first flip is a head, and the remaining two flips can be either heads or tails. This gives us 1 * 2 * 2 = 4 outcomes.
- Scenario 2: The first flip is a tail, and the second or third flip is a head. This gives us 1 * 1 * 2 = 2 outcomes. However, we need to exclude the outcome where all three flips are tails (T-T-T), so we subtract 1. So, the total number of outcomes for Scenario 2 is 2 - 1 = 1 outcome.
Therefore, the total number of outcomes where at least one is a head is 4 + 1 = 5 outcomes.
Step 3: Determine the number of outcomes where exactly two heads are obtained. In this case, exactly two heads can occur in three possible ways: H-H-T, H-T-H, or T-H-H. The number of outcomes for each possibility is 1, so the total number of outcomes where exactly two heads are obtained is 3.
Step 4: Calculate the conditional probability. The probability of obtaining exactly two heads in three flips of a coin, given that at least one is a head, is the ratio of the number of outcomes where exactly two heads occur to the number of outcomes where at least one is a head:
P(2H | at least 1H) = Number of outcomes with exactly 2 heads / Number of outcomes with at least 1 head
= 3 / 5
= 0.6
So, the probability of obtaining exactly two heads in three flips of a coin, given that at least one is a head, is 0.6 or 60%.