Problem

A housing estate consists of 320 houses. A house on the estate is selected at random.

$D$ denotes the event ‘the house has a driveway’

$X$ denotes the event ‘no cars per household’

$Y$ denotes the event ‘exactly 1 car per household’

$Z$ denotes the event ‘exactly 2 cars per household’

The Venn diagram shows the number of households in each region for the events $D$, $X$, $Y$ and $Z$.

(a) Find (i) $P(D)$

(ii) $P(D\cap X')$

(iii) $P(D'\cup Z')$

Given that the house has a driveway,

(b) find the probability that there are exactly 2 cars in the household.

(c) Write down 2 of the events $D$, $X$, $Y$ and $Z$ that are mutually exclusive.

(d) Determine whether the events $D$ and $X$ are independent. Justify your answer.

(e) Define in the context of this question, the event

(i) $D'\cap Z$

(ii) $D\cap(X\cup Y)$