Workers from a certain company have a choice of two routes for travelling to work each day. The travel time in minutes using route \(A\) and route \(B\) have independent normal distributions. The means and standard deviations of these distributions are shown in the following table.
| Mean | Standard Deviation |
|---|
| Route \(A\) | \(45\) | \(7\) |
| Route \(B\) | \(53\) | \(5\) |
Given that the probability that the travel time using route \(A\) takes longer than \(k\) minutes is less than \(0.2\), find the least value of \(k\), giving your answer to the nearest minute.[2]
James and John both work at the company and leave for work at the same time every day. James takes route \(A\) and John takes route \(B\).
Find the probability that on a randomly chosen day, James and John arrive at the company more than \(5\) minutes apart.[3]
Find the probability that out of \(5\) randomly chosen days, James arrives at the company later than John on at most \(1\) day.[2]
When there is a traffic jam, it will increase the travel time using route \(A\) by \(5\) minutes and increase the travel time using route \(B\) by \(8\%\).
Peter uses route \(A\) for \(3\) occasions and route \(B\) for \(1\) occasion. Given that a traffic jam occurs on all occasions, find the probability that average travel time for the \(4\) occasions is between \(50\) minutes and \(1\) hour. You may assume that the travel time for all the occasions are independent of each other.[4]