The probabilities of two events $A$ and $B$ are given as $P\left(A\right)=0.8$ and $P\left(B\right)=0.7$. What is the minimum value of $P(A\cap B)$?

The correct option is C

$0.5$

Explanation for the correct option:

Given,

$$\begin{gathered} P\left(A\right)=0.8 \\ P\left(B\right)=0.7 \end{gathered}$$

We know that, $P\left(A\right)+P\left(B\right)-P(A\cap B)\le 1$

So, $0.8+0.7-P(A\cap B)\le 1$

$$\begin{gathered} \Rightarrow P(A\cap B)\ge 1.5–1 \\ \Rightarrow P(A\cap B)\ge 0.5 \end{gathered}$$

Hence the minimum value of $P(A\cap B)=0.5$

Hence, the correct answer is option (C), $0.5$.