ASRJC Binomial Distribution Tutorial Q12

A biscuit factory produces biscuits which are packed into its signature gift boxes. On average, \(3\%\) of these gift boxes packed are substandard.

1. Show that the probability that a randomly chosen batch of \(30\) gift boxes will contain at least \(2\) gift boxes which are substandard is \(0.227\), correct to \(3\) significant figures.[2]

The operations unit of the factory decided to pack \(30\) gift boxes into one carton before delivery to all its retail outlets.

2. Find the probability that out of \(10\) randomly chosen cartons, there are more than \(4\) cartons with at least \(2\) substandard gift boxes.[2]
3. Find the probability that out of \(10\) randomly chosen cartons, there are at least \(2\) substandard gift boxes.[2]

Every week, the owner of each retail outlet will place order with the factory.

4. Find the maximum number of cartons to be ordered so that the chance of having no substandard gift box is more than \(5\%\).[4]