2019 HCI P2 Q7

A cafe sells sandwiches in \(2\)sizes, “footlong” and “\(6\)-inch”. The lengths in inches of “footlong” loaves have the distribution \(\mathrm{N}\left( 12.2,0.04 \right)\) and the lengths in inches of “\(6\)-inch” loaves have the distribution \(\mathrm{N}\left( 6.1,0.02 \right)\).

1. Is a randomly chosen “footlong” loaf more likely to be less than \(12\) inches in length or a randomly chosen “\(6\)-inch” loaf more likely to be less than \(6\) inches in length?[2]
2. Find the probability that two randomly chosen “\(6\)-inch” loaves have total length more than one randomly chosen “footlong” loaf.[2]

Sue buys a “\(6\)-inch” sandwich \(3\) times a week.

3. Find the probability that Sue gets at most one sandwich that is less than \(6\) inches in length in a randomly chosen week.[2]
4. Given that Sue gets more than four sandwiches that are less than \(6\)inches in length in a randomly chosen \(4\)-week period, find the probability that she gets exactly one such sandwich in the first week.[3]