2022 MI P2 Q11

In this question you should, where applicable, state clearly all the distributions that you use, together with the values of the appropriate parameters.

The time spent by each customer at S Supermarket is denoted by the random variable \(X\). \(X\) follows a normal distribution with mean \(a\) minutes and standard deviation \(b\) minutes.

1. Given that \(\mathrm{P}\left( X>c \right)=0.3\), where \(c\) is a constant. Find
   1. \(\mathrm{P}\left( a < X < c \right)\),[1]
   2. \(\mathrm{P}\left( c < X < a+b \right)\).[2]

Use \(a=35\) and \(b=10\) for the rest of this question.

Alice, Barry and Christine are three customers at S Supermarket who do not know each other. Find the probability that

2. the time spent by Alice differs from the time spent by Barry by less than a minute,[3]
3. the total time spent by Barry and Christine exceeds twice the time spent by Alice by at least \(15\) minutes,[3]
4. the total time spent by Alice, Barry and Christine at S Supermarket is at most \(110\) minutes.[3]
5. If Barry and Christine were members of the same family, what can you say about the validity of your answers in parts (ii), (iii) and (iv)?[2]