Q.If $A$ and $B$ are two independent events, then the probability of occurrence of at least one of $A$ and $B$ is given by $1 - P(A')\,P(B')$.

The key idea is that "at least one" means the complement of "none", and independence lets us multiply the probabilities of the complements.

Step 1: "At least one of $A$ or $B$" is the complement of "neither $A$ nor $B$", i.e., $A' \cap B'$.

Step 2: For independent events $A$ and $B$, their complements $A'$ and $B'$ are also independent. …

Why this approach works

When you see "at least one" in probability, your first instinct should be to think of the complement. The event "at least one of $A$ or $B$ occurs" is the opposite of "neither $A$ nor $B$ occurs". That complement is $A' \cap B'$ — both events fail.

For any two events, the probability of at least one is:

The difficulty is that $P(A' \cap B')$ is not simply $P(A')P(B')$ unless the events are independent. But here, the problem explicitly tells us $A$ and $B$ are independent. Independence of $A$ and $B$ also implies independence of their complements — a key fact that makes the formula work.

Step-by-step reasoning

1. State what we want.

   We need $P(\text{at least one of } A \text{ or } B)$. This is $P(A \cup B)$.

2. Use the complement rule.

3. Apply independence of complements. Since $A$ and $B$ are independent, $A'$ and $B'$ are independent. Therefore: $P(A' \cap B') = P(A') \cdot P(B')$ …

Method: The Complement Rule for "At Least One"

Use this whenever a probability question asks for "at least one" of several events occurring — it is almost always faster to work with the opposite event ("none occur") than to add up every "one-or-more" case.

Steps

Step 1: Rewrite "at least one" as the complement of "none".

The event "at least one of the given events occurs" is the exact opposite of "none of them occurs". For two events $A,B$,

Step 2: Evaluate $P(A' \cap B')$ using the structure of the events.

"Neither occurs" means both events fail together, i.e. $A' \cap B'$. In general this needs the joint probability, but if $A$ and $B$ are independent, then their complements $A'$ and $B'$ are also independent, so the joint factorises:

$P(A' \cap B') = P(A')\,P(B').$ …

Common Mistakes

Mistake 1: Computing "at least one" as $P(A)+P(B)$.

Why it's wrong: this double-counts the overlap $A\cap B$ and can even exceed $1$. Correct approach: use the complement, $1-P(A'\cap B')$, or the addition rule $P(A)+P(B)-P(A\cap B)$.

Mistake 2: Writing $P(A'\cap B')=P(A')P(B')$ without justifying independence.

Why it's wrong: the joint of the complements factorises only because $A,B$ are independent. Correct approach: state that independence of $A,B$ implies independence of $A',B'$, and only then multiply. …