Given events A and B, their probabilities are:
[tex]\begin{gathered} P(A)=0.4 \\ P(B)=0.5 \end{gathered}[/tex]
a.
If the events are mutually exclusive, this means that they can not occur at the same time. Then:
[tex]\begin{gathered} P(A\text{ and }B)=0...(1) \\ P(A\text{ or }B)=P(A)+P(B)...(2) \end{gathered}[/tex]
Additionally, from the formula for conditional probabilities, we have:
[tex]P(A|B)=\frac{P(A\text{ and }B)}{P(B)}...(3)[/tex]
Then, using the values for P(A), P(B), and (1):
[tex]\begin{gathered} P(A\text{ and }B)=0 \\ \\ P(A\text{ or }B)=0.4+0.5=0.9 \\ \\ P(A|B)=\frac{0}{0.5}=0 \end{gathered}[/tex]
b.
If the events are independent, this means that:
[tex]\begin{gathered} P(A\text{ and }B)=P(A)\cdot P(B)...(4) \\ \\ P(A|B)=P(A)...(5) \\ \\ P(A\text{ or }B)=P(A)+P(B)-P(A)\cdot P(B)...(6) \end{gathered}[/tex]
Finally, using P(A) and P(B) in (4), (5), and (6):
[tex]\begin{gathered} P(A\text{ and }B)=0.4\cdot0.5=0.2 \\ \\ P(A\text{ or }B)=0.4+0.5-0.4\cdot0.5=0.7 \\ \\ P(A|B)=0.4 \end{gathered}[/tex]
