Normal parameters from two tail percentages, binomial turn-up numbers with a conditional probability and the largest number of bookings

The time, \(T\) minutes, that a guided walking tour lasts follows a normal distribution, \(T\sim N(\mu,\sigma^2)\). Records show that \(4.5\%\) of tours last less than \(85\) minutes and \(9.2\%\) of tours last more than \(110\) minutes.

1. [7]
   1. Find the value of \(\mu\) and the value of \(\sigma\).
   2. Hence find, for a tour chosen at random, the probability that it lasts between \(90\) and \(100\) minutes.
2. Each person who books a place on a tour turns up with probability \(0.85\), independently of the other people who have booked.
   For one tour, \(25\) people have booked a place.[7]
   1. Find the probability that \(20\) or more of these people turn up.
   2. Given that \(20\) or more of these people turn up, find the probability that \(23\) or more of them turn up.
3. The tour minibus has \(36\) passenger seats. The company accepts more bookings than there are seats, but it requires the probability that every person who turns up has a seat to be no less than \(0.95\).
   Find the largest number of bookings that the company can accept for one tour.[4]