In this question, you should state the parameters of any distributions you use.
A company produces wooden planks of two different lengths, Regular and Long. The lengths, in metres, of the Long planks follow the distribution \(\mathrm{N}(1.82,0.2^2)\).
Find the probability that the length of a randomly chosen Long plank is less than \(1.79\text{ m}\).[1]
Find the probability that the total length of 8 randomly chosen Long planks is greater than \(14.5\text{ m}\).[3]
The lengths, in metres, of the Regular planks follow the distribution \(\mathrm{N}(1.22,0.3^2)\).
Sylvio buys 120 of the Regular planks. Calculate the expected number of these planks that are longer than \(1.25\text{ m}\).[2]
Find the probability that the total length of 10 randomly chosen Long planks differs by less than \(0.65\text{ m}\) from the total length of 16 randomly chosen Regular planks.[3]
The company finds that there is a demand for Short planks. These planks are produced by cutting the Long planks into three exactly equal lengths, or the Regular planks into two exactly equal lengths.
Find the probability that the length of a randomly chosen Short plank made from a Long plank is greater than the length of one made from a Regular plank. You should ignore any wastage caused by cutting the plank.[4]
Without doing any detailed calculation, explain how your answer to part (e) would change if each cut of a plank caused a small amount of wastage.[1]