2023 EJC P2 Q8

A company produces ceramic vases. A fixed number of randomly chosen vases are inspected each day and the number of defective vases found in a day is denoted by \(X\).

1. State, in context, two assumptions needed for \(X\) to be well modelled by a binomial distribution.[2]

Assume now that \(X\) has the distribution \(\mathrm{B}\left( 30,0.04 \right)\).

2. Find the probability that, on a randomly chosen day, more than \(2\) defective vases are found.[2]

The number of defective vases found each day is independent of that on other days.

3. Find the probability that, in a randomly chosen \(5\)-day working week, more than \(2\) defective vases are found on at most \(1\) day.[2]
4. In a particular \(5\)-day working week, a total of \(5\) defective vases were found. Find the probability that all \(5\) of them were found on exactly \(2\) days that are consecutive.[4]