Solving for Parameters, Comparisons, and Multi-Stage Distribution Models

A fruit grower grows both red and green apples which have masses that are normally distributed. The mass of a randomly chosen red apple has mean \(75\)g and standard deviation \(12.5\)g. The mass of a randomly chosen green apple has mean of \(55\)g and standard deviation \(10.5\)g.

1. Find the probability that the total mass of \(3\) randomly chosen green apples exceeds twice the mean mass of \(3\) randomly chosen red apples.[3]
2. A red apple is considered “underweight” if it weighs less than \(70\)g. Red apples are packed into bags of \(10\) for transportation to a supermarket. A bag is considered to have passed the quality test if it contains less than \(2\) “underweight” apples. Calculate the probability that in a randomly chosen batch of \(20\) bags of red apples, all the bags fail the quality test. [4]