RI Normal Distribution Tutorial Q8

In this question you should assume that \(T\), \(W\) and \(D\) follow independent normal distributions.

James leaves home to go to work at \(T\) minutes past 8 am each day, where \(T\) follows the distribution \(N\left( 5,\ {{1.2}^{2}} \right)\).

1. Sketch this distribution for the period from 8 am to 8.10 am.[2]
2. Find the probability that, on a randomly chosen day, James leaves for work later than 8.06 am.[1]

When the weather is fine, James walks to work. The time, \(W\) minutes, he takes to walk to work follows the distribution \(N\left( 21,\ {{3}^{2}} \right)\). James is supposed to start work at 8.30 am.

3. Find the probability that, on a randomly chosen day when James walks, he is late for work.[2]

When the weather is not fine, James drives to work. He still leaves at \(T\) minutes past 8 am each day; the time, \(D\) minutes, he takes to drive to work follows the distribution \(N\left( 19,\ {{6}^{2}} \right)\).

On average, the weather is fine on \(70\%\) of mornings.

4. One day, James is late for work. Find the probability that the weather is fine that day.[5]