Venn diagram, gold and Bitcoin

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In the Venn diagram, event $A$ is “tomorrow the gold price rises”, event $B$ is “tomorrow the Bitcoin price rises”. The regions are: $A$-only $=x$, $B$-only $=0.2$, intersection $A\cap B=0.3$, outside $=t$.

• (a) You know that $P(A)=40\%$. Find $x$ and $t$.
• (b) Find the probability that tomorrow gold does not rise.
• (c) Find the probability that gold rises given that Bitcoin also rises.
• (d) Decide whether the rise of gold and the rise of Bitcoin are independent events; decide whether they are mutually exclusive.

Solution

(a) $P(A)=x+0.3=0.4\Rightarrow x=0.1$. Sum to $1$: $0.1+0.2+0.3+t=1\Rightarrow t=0.4$.

(b) $P(\neg A)=1-P(A)=1-0.4=0.6$.

(c) $P(B)=0.2+0.3=0.5$, hence $P(A\mid B)=\frac{P(A\cap B)}{P(B)}=\frac{0.3}{0.5}=0.6.$

(d) Independence: $P(A)\,P(B)=0.4\cdot 0.5=0.2$, whereas $P(A\cap B)=0.3\ne 0.2$: the events are not independent (positively correlated). They are not mutually exclusive, since $P(A\cap B)=0.3\ne 0$.

$\boxed{x=0.1,\ t=0.4;\ P(\neg A)=0.6;\ P(A\mid B)=0.6;\ \text{dependent, not exclusive}}$