Answer:
(i) P(B) = 0.12
(ii) P(B) = 0.2
Explanation:
A bar over a set means that we should take the complement of that set. It can also be notated by an apostrophe:

A complement of a set refers to the elements that are not included in the set, but are part of the universal set.
The symbol "∪" means the union of sets. It represents the set that contains all the elements that are in either set or in both sets.
P(A ∪ B) represents the probability of the union of sets A and B, which is the event that either A or B or both occur. Therefore, P(A ∪ B)' represents the probability of the complement of P(A ∪ B), so the probability of the event that neither A nor B occurs. Mathematically, it can be defined as:


Part (i)
Mutually exclusive events are those that have no common outcomes and therefore cannot occur simultaneously. When represented using a Venn diagram, mutually exclusive events are depicted as non-overlapping circles.
The addition law for mutually exclusive events is:

Therefore, as P(A ∪ B)' = 1 - P(A ∪ B), we can say that:
![\begin{aligned} \sf P(A \cup B)'&=\sf 1-P(A \cup B)\\ &=\sf 1-[P(A)+P(B)]\end{aligned}](https://img.qammunity.org/2024/formulas/mathematics/high-school/m3hvdklvtczaieat5jd51wedetzehubemw.png)
Given P(A ∪ B)' = 0.48 and P(A) =0.4, substitute these into 1 - [P(A) + P(B)] and solve for P(B):
![\begin{aligned}\sf 1-[0.4+P(B)]&=\sf0.48\\\sf1-0.4-P(B)&=\sf0.48\\\sf 1-0.4-0.48&=\sf P(B)\\\sf P(B)&=\sf0.12\end{aligned}](https://img.qammunity.org/2024/formulas/mathematics/high-school/ybcbu5r86wnxs880g3e53ybriaymudbbkq.png)
Therefore, P(B) = 0.12 if events A and B are mutually exclusive.

Part (ii)
If the probability of an event B happening doesn’t depend on whether an event A has happened or not, events A and B are independent.
The addition law for independent events is:

The product law for independent events is

Therefore, as P(A ∪ B)' = 1 - P(A ∪ B), we can say that:
![\begin{aligned} \sf P(A \cup B)'&=\sf 1-P(A \cup B)\\ &=\sf 1-[P(A)+P(B)-P(A \cap B)]\\&=\sf 1-[P(A)+P(B)-P(A)P(B)]\end{aligned}](https://img.qammunity.org/2024/formulas/mathematics/high-school/cbcbp32ze0qgc431slget0jb2itqna0t9e.png)
Given P(A ∪ B)' = 0.48 and P(A) =0.4, substitute these into the found expression, and solve for P(B):
![\begin{aligned}\sf 1-[0.4+P(B)-0.4P(B)]&=\sf0.48\\\sf 1-[0.4+0.6P(B)]&=\sf 0.48\\\sf 1-0.4-0.6P(B)&=\sf 0.48\\\sf 0.6-0.6P(B)&=\sf 0.48\\\sf 0.6P(B)&=\sf 0.12\\\sf P(B)&=\sf 0.2\end{aligned}](https://img.qammunity.org/2024/formulas/mathematics/high-school/sr8r5bh6d3y5iant7tt8o3j6fhzctp300b.png)
Therefore, P(B) = 0.2 if events A and B are independent.