NYJC Tutorial Q2

A sample of \(6\) objects is to be drawn from a population in which \(20%\) are defective. Explain why, if there are \(10\,000\) objects in the whole population, the probability of obtaining \(3\) defective objects in the sample can be very accurately estimated by using the binomial probability formula, whereas, if there are only \(20\) objects in the population, this method gives quite the wrong answer.

For the large population, use this method to estimate the probability that a sample of \(6\) contains \(3\) or fewer defective objects.[ \(0.983 \)]

Another large population contains a proportion \(p\) of defective items. Write down an expression in terms of \(p\) for \(P\), the probability that a sample of \(6\) contains exactly \(2\) defectives. Find \(\frac{\mathrm{d}P}{\mathrm{d}p}\) and show that \(P\) is greatest when \(p=\frac{1}{3}\).