2021 MI P2 Q11

In this question, you should assume that \(L\) and \(S\) follow independent normal distributions. You should also state clearly the mean and variance distributions you use.

An oil company supplies engine oil in cans of two capacities, large and small.

The amount, \(L\) milliliters, of oil in a large can is normally distributed, where \(L\sim\mathrm{N}\left( 5000,{{\sigma }^{2}} \right)\). The amount, \(S\) milliliters, of oil in a small can is normally distributed, where \(S\sim\mathrm{N}\left( 1000,25 \right)\).

1. Find \(\mathrm{P}\left( 5000-\sigma < L < 5000+2\sigma \right)\).[2]
2. The probability that a randomly chosen large can has more than \(4990\) milliliters of oil is \(\mathrm{0}\mathrm{.943}\). Find \({{\sigma }^{2}}\).[2]

Use \({{\sigma }^{2}}=40\) for the rest of the question.

3. Find the probability that the amount of oil in a randomly chosen small can is at most \(\mathrm{1002}\) milliliters.[1]
4. Twenty small cans of oil are randomly chosen. Find the probability that fewer than ten cans have the amount of oil in the can to be at most \(1002\) milliliters.[2]
5. Find the probability that the amount of oil in a randomly chosen large can exceeds five times the amount of oil in a randomly chosen small can by more than \(30\) milliliters.[3]

The oil company fills the large cans with Type \(A\) lubricating oil and small cans with Type \(B\) lubricating oil, and sells the Type \(A\) industrial lubricating oil at \(\$0.13\) per milliliters and Type \(B\) industrial lubricating oil at \(\$0.05\) per milliliters. The oil company supplies \(6\) large cans of oil and \(2\) small cans of oil to a manufacturing firm.

6. Find the probability that the total cost incurred for the manufacturing firm is at least \(\$3995\).[4]