A batch of one hundred products is subjected to partial…

A batch of one hundred products is subjected to partial inspection: five products are inspected, and if at least one defective product appears among them, the whole batch is rejected. What is the probability that the batch will be rejected if it contains $5\%$ defective products?

Step 1: The batch

$5\%$ of $100$: $5$ defective and $95$ good products. Five products are chosen without replacement.

Step 2: The opposite event — all five good

By counting: $P(\text{all good})=\dfrac{\binom{95}5}{\binom{100}5}$. Draw by draw (multiplication rule with conditional probabilities):

\[ P(\text{all good}) = \frac{95}{100}\cdot\frac{94}{99}\cdot\frac{93}{98}\cdot\frac{92}{97}\cdot\frac{91}{96}\approx 0.7696. \]

Step 3: The answer

\[ \boxed{P(\text{rejected}) = 1-\frac{\binom{95}5}{\binom{100}5}\approx 0.2304.} \]

Step 4: Approximation with replacement

If the inspected items were returned (or the batch were very large), each would be good with probability $0.95$ independently: $P\approx1-0.95^5\approx0.2262$. This is close, but not exact: without replacement, after drawing good items the share of defective ones among the rest increases slightly, so rejection is a little more likely ($0.2304\gt0.2262$).

Step 5: Interpretation

Even with $5\%$ defective products, the inspection of only five items rejects the batch less than one time in four. To make the check reliable, more items must be inspected.