The probability that a vehicle will fail an emissions test by the Department of Motor Vehicles is 15% (0.15). If 12 vehicles undergo the emissions test, what is the probability that 3 vehicles will fail the test?  Assume a Binomial Distribution


Do not answer this question with a percentage

We are given that the probability of failure for a vehicle during the emissions test is \( p = 0.15 \), and the number of vehicles undergoing the test is \( n = 12 \). We are asked to find the probability that exactly 3 vehicles will fail the test, which follows a binomial distribution.

The binomial probability formula is:

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

Where:
- \( n \) is the number of trials (in this case, 12 vehicles),
- \( k \) is the number of successes (here, 3 vehicles failing the test),
- \( p \) is the probability of success (here, failure) on a single trial (0.15),
- \( \binom{n}{k} \) is the binomial coefficient, calculated as \( \frac{n!}{k!(n - k)!} \).

Let’s calculate:

### Step 1: Calculate the binomial coefficient \( \binom{12}{3} \)
\[
\binom{12}{3} = \frac{12!}{3!(12 - 3)!} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220
\]

### Step 2: Calculate \( p^k = (0.15)^3 \)
\[
p^3 = (0.15)^3 = 0.003375
\]

### Step 3: Calculate \( (1 - p)^{n - k} = (0.85)^9 \)
\[
(0.85)^9 \approx 0.272490525
\]

### Step 4: Putting everything together
\[
P(X = 3) = 220 \times 0.003375 \times 0.272490525
\]
\[
P(X = 3) \approx 0.202
\]

Thus, the probability that exactly 3 vehicles will fail the test is approximately **0.202**.

Would you like a detailed explanation of any step?

Here are 5 questions related to this topic:
1. How is the binomial distribution formula derived?
2. What is the significance of the binomial coefficient \( \binom{n}{k} \)?
3. How does the probability change if more vehicles are tested?
4. What is the difference between binomial and normal distributions?
5. How can the binomial distribution be approximated with a normal distribution?

**Tip**: In problems involving binomial distributions, recognizing the values for \( n \), \( p \), and \( k \) is essential before plugging them into the formula.

The probability that a vehicle will fail an emissions test by the Department of Motor Vehicles is 15% (0.15). If 12 vehicles undergo the emissions test, what is the probability that 3 vehicles will fail the test? Assume a Binomial Distribution.

Find out how to calculate the probability of exactly 3 vehicles failing an emissions test when the failure rate is 15%. This solution explains the use of the binomial distribution formula, covering probability concepts suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation