2025 CJC P2 Q10

Fishing Company A has two types of fishing vessels, the “Standard” vessel and the “Large” vessel. The amount of fish caught by the respective vessels on a typical fishing trip are normally distributed with means and standard deviations as shown in the table.

1. If a Standard vessel returns with more than \(400\) kg of fish, it is called a Bumper catch. Bumper catches occur \(8\%\) of the time. Show that \(\sigma \approx 71.2\).[2]
2. On a particular fishing trip, Company A sends out \(3\) Standard vessels and \(2\) Large vessels, with a target to catch at least \(2100\) kg of fish. Find the probability that the fishing vessels are able to meet their target.[2]
3. State an assumption needed in your calculation in part (b).[1]

Fishing Company B operates in the same seas as Fishing Company A. However, Company B has the improved versions of the two types of fishing vessels that Company A has: the “Premium-Standard” vessel and the “Premium-Large” vessel. In a fishing trip, the Premium-Standard vessel is capable of catching \(3\) times as much fish as the Standard vessel and the Premium-Large vessel is capable of catching \(2\) times as much fish as Large vessel.

4. Company B sends out \(1\) Premium-Standard vessel and \(1\) Premium-Large vessel, with the same target to catch at least \(2100\) kg of fish. Find the probability that these two fishing vessels are able to meet their target.[2]
5. It is found that, in a sample of \(n\) randomly chosen fishing trips, the probability that the average catch of a Large vessel being less than \(552\) kg is at least \(0.7\). Find the least possible value of \(n\).[3]