In this situation, the binomial probability distribution can be used to model the probability of successfully making a basketball shot. If the probability of making a shot is 60%, then the probability of making 5 shots out of 12 attempts would be calculated as follows:
1-0.60 = 0.40 of missing the free throw 
n=12 , x =7, p= 0.60 , q= 0.40


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.

In this situation, the binomial probability distribution can be used to model the probability of successfully making a basketball shot. If the probability of making a shot is 60%, then the probability of making 5 shots out of 12 attempts would be calculated as follows: 1-0.60 = 0.40 of missing the free throw, n=12, x=7, p=0.60, q=0.40.

Learn how to calculate the probability of making exactly 7 shots out of 12 basketball attempts with a 60% success rate using the binomial distribution formula. This step-by-step guide covers key probability concepts, including the binomial coefficient and binomial theorem, and is suitable for high school students (Grades 10-12).

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