Venn diagram, temperature and rain

Text

In the Venn diagram, event $A$ is “tomorrow the temperature exceeds $30$ degrees”, event $B$ is “tomorrow it rains”. The regions are: $A$-only $=x$, $B$-only $=0.2$, intersection $A\cap B=0.3$, outside $=t$.

• (a) You know that the probability that it rains tomorrow given that it is above $30$ degrees is $\tfrac12$. Find $x$ and $t$, and prove that $x=0.3$.
• (b) Find the probability that tomorrow it does not rain.
• (c) Find the probability that tomorrow it does not rain given that it is below $30$ degrees.
• (d) Decide whether the two events $A$ and $B$ are independent.

(In the original test the value in part (a) is written $3/8$: this is a typo — with $3/8$ one would get $x=0.5$ and $t=0$, incompatible with a proper diagram. The value giving $x=0.3$, as the text itself requires, is $P(B\mid A)=\tfrac12$.)

Solution

(a) $P(B\mid A)=\dfrac{P(A\cap B)}{P(A)}=\dfrac{0.3}{x+0.3}=\dfrac12\Rightarrow x+0.3=0.6\Rightarrow x=0.3$. Sum to $1$: $0.3+0.2+0.3+t=1\Rightarrow t=0.2$.

(b) $P(\neg B)=P(B\text{-only})+P(\text{outside})=x+t=0.3+0.2=0.5$.

(c) $\neg A$ (below $30$ degrees) has probability $0.2+0.2=0.4$; $\neg A\cap\neg B$ is the outside $=0.2$: $P(\neg B\mid\neg A)=\frac{0.2}{0.4}=0.5.$

(d) $P(A)=0.3+0.3=0.6$, $P(B)=0.2+0.3=0.5$, hence $P(A)\,P(B)=0.30=P(A\cap B)$: the events are independent (indeed $P(B\mid A)=P(B)=0.5$, consistent with part (c)).

$\boxed{x=0.3,\ t=0.2;\ P(\neg B)=0.5;\ P(\neg B\mid\neg A)=0.5;\ \text{independent}}$