Records had shown that $18\%$ of the eggs from a particular farm were classified as “small” while

Question

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, $m<20$. 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. Find the probability that the batch is eventually accepted. Mr Egg randomly selects $60$ eggs from the farm. Find the probability that the total number of “small” eggs and “medium” eggs in his sample is more than $55$.

Accepted Answer

(i) $m=18.6$ (ii)$0.710$(iii) $0.470$

2013 RI P2 Q11 [Modified]

2013 RI P2 Q11 [Modified]