Q.Let A and B be independent events with $P(A) = 0.3$ and $P(B) = 0.4$. Find

Concept: Event Independence — For independent events, the occurrence of one does not affect the other, so $P(A \cap B) = P(A) \cdot P(B)$.

Step 1: Compute $P(A \cap B)$ using the product rule for independence:

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

Step 2: Use the addition rule for $P(A \cup B)$:

$P(A \cup B) = P(A) + P(B) - P(A \cap B) = 0.3 + 0.4 - 0.12 = 0.58$. …

The core idea here is event independence. When two events are independent, the occurrence of one has no effect on the probability of the other. This is the defining property, and it gives us a clean, direct way to compute everything.

Why independence matters:

If $A$ and $B$ are independent, then $P(A \cap B) = P(A) \cdot P(B)$. This is not a theorem we prove from something else — it is the definition of independence for two events. From this single fact, all other results follow naturally. The conditional probabilities become trivial: $P(A|B) = P(A)$ and $P(B|A) = P(B)$, because knowing $B$ happened gives no new information about $A$, and vice versa.

Let’s work through each part step by step.

1. Find $P(A \cap B)$ Since $A$ and $B$ are independent, the probability that both occur is simply the product of their individual probabilities.

2. Find $P(A \cup B)$ The union probability is given by the inclusion-exclusion principle:

We already have $P(A \cap B) = 0.12$, so:

3. Find $P(A|B)$ The conditional probability of $A$ given $B$ is defined as:

Substituting the values:

$P(A|B) = \frac{0.12}{0.4} = 0.3$ …

Method: Computing Derived Probabilities for Independent Events

Use this when two events are given as independent and you must find several derived quantities — the intersection, the union, and the conditionals.

Steps

Step 1: Intersection via the product rule

Independence gives the joint probability directly:

Step 2: Union via the addition rule

Feed that intersection into

Don't skip the subtraction — that shortcut only holds for mutually exclusive events. …

Common Mistakes

Mistake 1: Forgetting to subtract the intersection in $P(A\cup B)$.

Writing $P(A\cup B)=0.3+0.4=0.7$ double-counts the overlap. Why it's wrong: that sum-only formula holds only for mutually exclusive events. Correct approach: $P(A\cup B)=P(A)+P(B)-P(A\cap B)=0.3+0.4-0.12=0.58$.

Mistake 2: Over-working (or mis-reading) the conditional probabilities. …