(d) Neil throws three darts, what is the probability that he hits 20 exacthy twice?

To solve the problem, we need to determine the probability that Neil hits the score of 20 exactly twice when he throws three darts. ### Known Conditions: 1. Neil throws three darts. 2. We want the probability of hitting 20 exactly twice. ### Steps to Solve: 1. **Define the Probability of Hitting 20**: Let \( p \) be the probability of hitting 20 with a single dart throw. The probability of not hitting 20 is \( q = 1 - p \). 2. **Determine the Number of Successful Outcomes**: We need to find the number of ways to hit 20 exactly twice in three throws. This can be calculated using the binomial coefficient: \[ \text{Number of ways} = \binom{n}{k} = \binom{3}{2} \] where \( n \) is the total number of throws (3) and \( k \) is the number of successful hits (2). 3. **Calculate the Probability**: The probability of hitting 20 exactly twice and missing once can be expressed as: \[ P(X = 2) = \binom{3}{2} p^2 q^1 \] 4. **Substituting Values**: Now we can substitute the values into the formula. Let's calculate the binomial coefficient and the probability expression. First, calculate the binomial coefficient: \[ \binom{3}{2} = 3 \] Now, the probability expression becomes: \[ P(X = 2) = 3 p^2 q \] ### Final Expression: Thus, the probability that Neil hits 20 exactly twice when throwing three darts is: \[ P(X = 2) = 3 p^2 (1 - p) \] If you have a specific value for \( p \), we can calculate the exact probability. Please provide the probability of hitting 20 with a single dart throw.