2013 RI P2 Q11 [Modified]

Records had shown that \(18\%\) of the eggs from a particular farm were classified as “small” while \(8\%\) of the eggs from the farm were classified as “large”. The rest were classified as “medium” eggs, of which \(m\%\) were substandard, \(m<20\).

1. Mrs Egg randomly selects \(10\) medium eggs from the farm. Given that the probability that Mrs Egg found \(2\) substandard medium eggs is \(0.3\), write down an equation involving \(m\) and hence find \(m\).
   A two-stage inspection scheme for accepting or rejecting a large batch of eggs from the farm is as follows:
   Stage 1: Randomly select and inspect a tray of \(30\) eggs.
   \(\,\,\,\,\,\,\,\,\,\,\,\) Accept the batch if the tray contains at most \(5\) “small” eggs.
   \(\,\,\,\,\,\,\,\,\,\,\,\) Proceed to stage 2 if the tray contains \(6\) or \(7\) “small” eggs.
   \(\,\,\,\,\,\,\,\,\,\,\,\) Reject the batch otherwise.
   
   Stage 2: Randomly select and inspect a further sample of \(10\) eggs.
   \(\,\,\,\,\,\,\,\,\,\,\,\) Accept the batch if the further sample contains at least \(1\) “large” egg.
   \(\,\,\,\,\,\,\,\,\,\,\,\) Reject the batch otherwise.
2. Find the probability that the batch is eventually accepted.

Mr Egg randomly selects \(60\) eggs from the farm.

3. Find the probability that the total number of “small” eggs and “medium” eggs in his sample is more than \(55\).