Let A and B be independent events with P(A)=0.3 and P(B)=0.4. Find (i) P(A \cap B)
(ii) P(A \cup B)
(iii) P(A \mid B)
(iv) P(B \mid A)
Let A and B be independent events with P(A)=0.3 and P(B)=0.4. Find (i) P(A \cap B)
(ii) P(A \cup B)
(iii) P(A \mid B)
(iv) P(B \mid A)

Given P ( A )=0.3 and P ( B )=0.4

(i) P(A \cap B)

When A and B are independent. Two events are independent, statistically independent, or stochastically independent if the occurrence of one does not affect the probability of occurrence of the other.

\Rightarrow P(A\cap B)=P(A)\cdot P(B)

\Rightarrow P(A\cap B)=0.3\times 0.4\Rightarrow P(A\cap B)=0.12

(ii)

As we know,

    \[P(A \cup B)=P(A)+P(B)-P(A \cap B)\]

    \[\Rightarrow P(A \cup B)=0.3+0.4-0.12\]

    \[\Rightarrow P(A \cup B)=0.58\]

(iii) P ( A \mid B )

As we know

    \[P(A \mid B)=\frac{P(A \cap B)}{P(B)}\]

    \[P(A\mid B)=\frac{0.12}{0.4}\Rightarrow P(A\mid B)=0.3\]

(iv)

As we know

    \[P(B \mid A)=\frac{P(A \cap B)}{P(A)}\]

    \[\Rightarrow P(B\mid A)=\frac{0.12}{0.3}\Rightarrow P(B\mid A)=0.4\]