2019 TJC BT2 P2 Q10 [Modified]

Beng and Charlie are avid amateur farmers who specialised in growing carrots, which are then sold to the local grocer. Beng claims that the lengths of his carrots are normally distributed with mean \(22\) cm and standard deviation \(14\) cm, while Charlie claims that the lengths of his carrots are normally distributed with mean \(19\) cm and standard deviation \(6\) cm.

Explain why Beng’s claim is not likely to be true, supporting your answer numerically.[2]

Assuming Charlie’s claim is true.

1. \(3\) carrots from Charlie’s harvest are randomly selected. Find the probability that the total length of first two randomly selected carrots from Charlie’s harvest is shorter than twice the length of the third randomly selected carrot from Charlie’s harvest by at least \(1\) cm.[3]
2. It costs Charlie an average of \(6\) cents to grow a carrot, and he hopes to sell his carrots to the grocer at \(9\) cents each. However, the grocer will only buy carrots that are at least \(l\) cm long. Find the maximum value of \(l\), where \(l\) is an integer, for which Charlie can be expected to make a profit off his carrots.[3]

The grocer then decides to buy Charlie’s carrots for \(9\) cents each if they are at least \(13\) cm long. Given that Charlie sells his harvest of \(14\) carrots and makes a profit, find the probability that at most \(2\) of Charlie’s carrots are shorter than \(13\) cm in length.[4]