Let's break the question into parts and solve each one systematically:

QUESTION FOUR (20 MARKS)

(a) Define the following terms as used in probability and statistics:

(i) Sample space (1 Mark)
The sample space is the set of all possible outcomes of a probability experiment.
For example, in tossing a coin, the sample space is $S = \{ \text{Heads, Tails} \}$.

(ii) Event (1 Mark)
An event is any subset of the sample space, representing one or more outcomes of a probability experiment.
For example, in rolling a die, the event "rolling an even number" is $E = \{2, 4, 6\}$.

(iii) Mutually exclusive event (2 Marks)
Two events are mutually exclusive if they cannot occur at the same time (i.e., they have no common outcomes).
For example, in tossing a coin, the events "Heads" and "Tails" are mutually exclusive because you cannot get both outcomes in a single toss.

(b) Two events A and B are independent. Given that $P(A) = 0.3$ and $P(B) = 0.4$, find:

1. $P(A \cap B)$ (2 Marks)
   Since $A$ and $B$ are independent:
   $P(A \cap B) = P(A) \cdot P(B)$
   Substituting the values:
   $P(A \cap B) = 0.3 \cdot 0.4 = 0.12$

2. $P(A \cup B)$ (2 Marks)
   The formula for the union of two events is:
   $P(A \cup B) = P(A) + P(B) - P(A \cap B)$
   Substituting the values:
   $P(A \cup B) = 0.3 + 0.4 - 0.12 = 0.58$

Would you like a deeper explanation of these solutions or clarifications on any part? Here are some related questions you can explore:

Related Questions

1. How do you calculate $P(A \cap B)$ for dependent events?
2. What is the relationship between mutually exclusive events and independent events?
3. How does the sample space change if the number of trials in an experiment increases?
4. Can an event be both independent and mutually exclusive? Why or why not?
5. What is the complement of an event, and how is its probability calculated?

Tip:

When solving probability problems, always start by identifying whether the events are independent, mutually exclusive, or overlapping, as this determines the formula to use.