Quality Control
Suppose that a company selects two people who work independently inspecting two-by-four timbers. Their job is to identify low-quality timbers. Suppose that the probability that an inspector does not identify a low-quality timber is 0.20.
b. How many inspectors should be hired to keep the probability of not identifying a low-quality timber below 1%?

Verified step by step guidance

1. First, define the probability that a single inspector does not identify a low-quality timber as \(p = 0.20\).
2. Since the inspectors work independently, the probability that none of the \(n\) inspectors identify the low-quality timber is \(p^n = (0.20)^n\).
3. We want to find the smallest number of inspectors \(n\) such that the probability of not identifying the low-quality timber is less than 1%, or mathematically, \(p^n < 0.01\).
4. To solve for \(n\), take the natural logarithm of both sides: \(\ln(p^n) < \ln(0.01)\), which simplifies to \(n \ln(p) < \ln(0.01)\).
5. Finally, solve for \(n\) by dividing both sides by \(\ln(p)\) (noting that \(\ln(p)\) is negative since \(p < 1\)), giving \(n > \frac{\ln(0.01)}{\ln(0.20)}\). The smallest integer \(n\) satisfying this inequality is the number of inspectors needed.

Key Concepts

Probability of Independent Events

When events are independent, the probability of all events occurring together is the product of their individual probabilities. In this context, each inspector's failure to identify a low-quality timber is independent, so the combined probability of all inspectors missing it is the product of their individual failure probabilities.

Complement Rule in Probability

The complement rule states that the probability of an event not happening is one minus the probability that it does happen. Here, the goal is to keep the probability of missing a low-quality timber below 1%, so we use complements to relate the success and failure probabilities of inspectors.

Exponential Decay of Failure Probability with Multiple Inspectors

As more independent inspectors are added, the probability that all fail to identify a low-quality timber decreases exponentially. This concept helps determine the minimum number of inspectors needed to reduce the overall failure probability below a specified threshold.