https://file.aiquickdraw.com/mathbot/file/1727069693829gcx7grap.jpg

To solve the problem in the image, we can model it using the **binomial probability formula** since we are dealing with independent events (whether a car passes or fails the emissions test) with only two possible outcomes.

The problem specifies that:
- The probability that a car fails the test (\(p\)) is 10%, or 0.10.
- There are 10 cars in total, so \(n = 10\).
- We are asked to find the probability that exactly 3 cars fail, so \(k = 3\).

The **binomial probability formula** is:
\[
P(X = k) = \binom{n}{k} \cdot p^k \cdot (1 - p)^{n - k}
\]
Where:
- \( \binom{n}{k} \) is the binomial coefficient, calculated as \( \frac{n!}{k!(n-k)!} \),
- \( p^k \) is the probability of exactly \(k\) successes (failures, in this case),
- \( (1 - p)^{n - k} \) is the probability of \(n - k\) failures (passes, in this case).

Now let's calculate this probability.

Find the probability that exactly three out of ten automobiles fail the emissions test, given that 10% of registered automobiles in Colorado fail the test.

This problem involves calculating the probability that exactly three cars fail an emissions test out of 10, using the binomial distribution formula. Given that 10% of cars in Colorado fail the test, this example provides a clear step-by-step solution for understanding binomial probability.

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