HardMathematicsHigh School (11–12)Statistics and ProbabilityNumericalConceptual

An inspector tests 8 items from a production line…

Try it on paper first. A hint, the key idea and the worked solution are here when you need them.

An inspector tests $8$ items from a production line. Each item is defective with the same probability $p$ , independently of the others. She notices that in her records, the probability of finding exactly $2$ defective items is the same as the probability of finding exactly $3$ . She wants to estimate the rate of defects from this fact, since her records do not show $p$ directly. Find $p$ . Find the mean number of defective items in a sample of $8$ . Find the probability of exactly $2$ defective items. The mean will help her plan how many spare items to keep, and the probability will be used in her report. Give decimals to four places. Use the ratio $\dfrac{P(X = 3)}{P(X = 2)}$ to find $p$ .

Need a reminder? The key ideaShow

For $n$ independent trials with success probability $p$ and $q = 1 - p$ , $P(X = k) = \binom{n}{k}p^k q^{n-k}$ , with mean $np$ and variance $npq$ . Check the four conditions first: fixed $n$ , two outcomes, the same $p$ , independent trials.

The worked solution

Have you tried it?

Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.

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