A tablet compression machine has a defect rate of 12%. What is the probability that if 10 tablets are produced, 8 will be defective?

To solve this problem, we will use the **binomial probability formula** since we are dealing with a fixed number of trials (in this case, 10 tablets), and each trial can either result in a defective or non-defective tablet.

The **binomial probability formula** is:

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

Where:
- \( P(X = k) \) is the probability of exactly \( k \) successes (in this case, defective tablets).
- \( \binom{n}{k} \) is the binomial coefficient (the number of ways to choose \( k \) successes from \( n \) trials).
- \( n \) is the total number of trials (in this case, 10 tablets).
- \( k \) is the number of successes (in this case, 8 defective tablets).
- \( p \) is the probability of success on a single trial (in this case, 0.12, the defect rate).
- \( 1 - p \) is the probability of failure on a single trial (in this case, 0.88, the non-defective rate).

Let's plug in the values:
- \( n = 10 \)
- \( k = 8 \)
- \( p = 0.12 \)

Now calculate the probability.The probability that exactly 8 out of 10 tablets produced will be defective, given a defect rate of 12%, is approximately \( 1.50 \times 10^{-6} \), or 0.0000015.

Would you like more details on the calculation? Here are some related questions:

1. What is the probability that 0 tablets are defective?
2. What is the probability that at least 2 tablets are defective?
3. What is the expected number of defective tablets in a batch of 10?
4. How does the defect rate change if the probability of defect increases to 20%?
5. What is the cumulative probability of producing 7 or fewer defective tablets?

**Tip**: For small probabilities like this, increasing the sample size might yield more meaningful insights.

Explore how to calculate the probability that exactly 8 out of 10 tablets produced by a machine will be defective, given a defect rate of 12%. This problem applies the binomial distribution formula and provides insights into probability theory. Suitable for advanced high school and college-level students.

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