9758/2020/P2/Q06

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 $\mathrm{N}\left(5,1.2^{2}\right)$

(i) Sketch this distribution for the period from $8 \mathrm{am}$ to $8.10 \mathrm{am}$.

[2]

(ii) 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 $\mathrm{N}\left(21,3^{2}\right)$. James is supposed to start work at $8.30 \mathrm{am}$.

(iii) 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 $\mathrm{N}\left(19,6^{2}\right)$.

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

(iv) One day, James is late for work. Find the probability that the weather is fine that day.

[5]