Q.If $A$ and $B$ are two independent events with $P(A) = \dfrac{3}{5}$ and $P(B) = \dfrac{4}{9}$, then $P(A' \cap B')$ equals
(A) $\dfrac{4}{15}$
(B) $\dfrac{8}{45}$
(C) $\dfrac{1}{3}$
(D) $\dfrac{2}{9}$

Concept: Event Independence — For independent events $A$ and $B$, the complements $A'$ and $B'$ are also independent.

Step 1:

$P(A') = 1 - P(A) = 1 - \frac{3}{5} = \frac{2}{5}$

$P(B') = 1 - P(B) = 1 - \frac{4}{9} = \frac{5}{9}$

Step 2:

Since $A'$ and $B'$ are independent,

$P(A' \cap B') = P(A') \cdot P(B')$ …

The core idea here is Event Independence. When two events are independent, knowing that one occurred gives you no information about whether the other occurred. This property extends beautifully to their complements: if $A$ and $B$ are independent, then $A'$ and $B'$ are also independent. Why? Because the logic of "no influence" works both ways — if $A$ doesn't affect $B$, then the absence of $A$ doesn't affect the absence of $B$ either.

The expression $P(A' \cap B')$ is the probability that neither $A$ nor $B$ happens. In set terms, this is the complement of the union: $A' \cap B' = (A \cup B)'$. But instead of going through the union, independence gives us a direct multiplication path.

Let's work it step by step.

1. Find the complements' probabilities. Since $P(A) = \frac{3}{5}$, we have

Similarly, $P(B) = \frac{4}{9}$, so

2. Apply independence to the complements. Because $A$ and $B$ are independent, $A'$ and $B'$ are also independent. Therefore,

3. Simplify the product. Multiply numerators and denominators:

Reduce by dividing numerator and denominator by 5:

$\frac{10}{45} = \frac{2}{9}.$ …

Method: Independence carries over to complements

Use this whenever events are stated independent and you need the probability that neither occurs.

Steps

Step 1: Use the key fact.

If $A$ and $B$ are independent, then so are $A'$ and $B'$ (and the mixed pairs $A,B'$ and $A',B$).

Step 2: Take complement marginals and multiply.

Step 3: Simplify. …

Common Mistakes

Mistake 1: Using $P(A'\cap B')=1-P(A\cap B)$.

Why it's wrong: that right side is $P\big((A\cap B)'\big)$ = "not both," which also counts the cases where exactly one event happens — a different event from "neither." Correct approach: $P(A'\cap B')=P(A')P(B')$ by independence of complements. …