To calculate the probability of making exactly 7 shots out of 12 attempts, we use the binomial probability formula:

$P(X = x) = \binom{n}{x} p^x q^{n-x}$

Where:

• $n = 12$ (number of trials or attempts),
• $x = 7$ (number of successful shots),
• $p = 0.60$ (probability of success, making a shot),
• $q = 0.40$ (probability of failure, missing a shot, $q = 1 - p$).

Now, let's plug these values into the formula.

Step 1: Calculate the binomial coefficient

The binomial coefficient $\binom{n}{x}$ is calculated as:

$\binom{n}{x} = \frac{n!}{x!(n-x)!} = \frac{12!}{7!(12-7)!} = \frac{12!}{7!5!}$

Step 2: Apply the binomial probability formula

The binomial probability formula becomes:

$P(X = 7) = \binom{12}{7} (0.60)^7 (0.40)^5$

I will calculate the exact result for you.The probability of making exactly 7 shots out of 12 attempts, given a 60% success rate, is approximately $0.227$ or 22.7%.

Would you like further details or explanations on any part of this? Here are some related questions for deeper exploration:

1. What if the success rate was 70% instead of 60%?
2. How would the probability change for making exactly 5 shots instead of 7?
3. How do you calculate the cumulative probability for making at least 7 shots?
4. What is the expected number of successful shots for 12 attempts with a 60% success rate?
5. Can we use a normal approximation to the binomial distribution for this scenario?

Tip: When the number of trials is large, the binomial distribution can often be approximated by a normal distribution, making calculations easier.