Question 10

The Venn diagram shows the probabilities associated with three events $X$ , $Y\,$ and $Z$ , representing students choosing to study different languages. The probability of students studying only $X\,$ is 0.15, only $Y\,$ is 0.2, and only $Z\,$ is 0.4. The intersection $\cap Y\,$ is $q\,$ and $\cap Z\,$ is $p$ . Events $X\,$ and $Z\,$ are disjoint. The probability of studying none of these languages is $r$ .

Venn diagram with events X and Z disjoint, each overlapping Y, showing probabilities 0.15, 0.2, 0.4, p, q and r in their respective regions.

Write down the two events that are mutually exclusive.

Given that $P (Y ∣ Z) = 0.2$ , find the value of $p$ .

Given also that $X\,$ and $Y\,$ are independent, find the values of $q\,$ and $r$ .