According to a recent DMV’s data, about 65% of the applicants can pass a driver’s proficiency road test. If you collect a random sample of 12 applicants, find the probability that exactly 7 applicants will pass.

This problem can be solved using the **binomial probability formula**, which is used when you have a fixed number of independent trials, each with two possible outcomes (success or failure). Here, passing the test is considered a "success."

The binomial probability formula is:

\[
P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}
\]

Where:
- \( n = 12 \) (the number of applicants),
- \( k = 7 \) (the number of applicants who pass),
- \( p = 0.65 \) (the probability that a single applicant passes),
- \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \) is the binomial coefficient.

### Step-by-step solution:

1. **Set up the formula:**
   \[
   P(X = 7) = \binom{12}{7} (0.65)^7 (0.35)^5
   \]

2. **Find the binomial coefficient** \( \binom{12}{7} \):
   \[
   \binom{12}{7} = \frac{12!}{7!(12-7)!} = \frac{12!}{7!5!} = 792
   \]

3. **Calculate the powers of the probabilities:**
   \[
   (0.65)^7 = 0.0493 \quad \text{and} \quad (0.35)^5 = 0.00525
   \]

4. **Multiply all the components together:**
   \[
   P(X = 7) = 792 \times 0.0493 \times 0.00525 = 0.205
   \]

Thus, the probability that exactly 7 out of 12 applicants will pass the test is approximately **0.205**, or **20.5%**.

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Would you like any further details or have any questions?

Here are some related questions:

1. What is the probability that more than 7 applicants will pass?
2. How would the probability change if the passing rate were 70% instead of 65%?
3. What is the probability that fewer than 7 applicants will pass?
4. How does the binomial distribution change with different sample sizes?
5. What is the expected number of applicants to pass out of 12?

**Tip:** In binomial problems, if you're dealing with large sample sizes, you can use the normal approximation to the binomial for faster computations.

Calculate the probability that exactly 7 out of 12 applicants will pass a driver's proficiency test with a 65% success rate. This step-by-step solution uses the binomial probability formula and covers core concepts in probability. Suitable for students in Grades 10-12.

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