Answer: The required probabilities are
[tex](a)~P(A/B)=\dfrac{1}{2},\\\\(b)~P(B/A)=\dfrac{2}{3}[/tex]
Step-by-step explanation: We are given the following probabilities for the events A and B :
[tex]P(A)=0.3,~~P(B)=0.4,~~\textup{and}~~P(A\cup B)=0.5.[/tex]
We are to find the values of the following probabilities :
[tex](a)~P(A/B),\\\\\\(b)~P(B/A).[/tex]
From the laws of probabilities, we have
[tex]P(A\cup B)=P(A)+P(B)-P(A\cap B)\\\\\Rightarrow P(A\cap B)=P(A)+P(B)-P(A\cup B)\\\\\Rightarrow P(A\cap B)=0.3+0.4-0.5\\\\\Rightarrow P(A\cap B)=0.2[/tex]
Therefore, we get
[tex]P(A/B)=\dfrac{P(A\cap B)}{P(B)}=\dfrac{0.2}{0.4}=\dfrac{1}{2},\\\\\\P(B/A)=\dfrac{P(B\cap A)}{P(A)}=\dfrac{0.2}{0.3}=\dfrac{2}{3}[/tex]
Thus, the required probabilities are
[tex](a)~P(A/B)=\dfrac{1}{2},\\\\\\(b)~P(B/A)=\dfrac{2}{3}.[/tex]
