Q.If $P(A) = \frac{3}{5}$ and $P(B) = \frac{1}{5}$, find $P(A \cap B)$ if A and B are independent events.

Concept: Independent Events — For independent events, the probability of both occurring is the product of their individual probabilities.

Step 1: Recall the definition of independence:

$P(A \cap B) = P(A) \cdot P(B)$.

Step 2: Substitute the given values:

$P(A \cap B) = \frac{3}{5} \times \frac{1}{5}$.

Step 3: Multiply:

$\frac{3}{5} \times \frac{1}{5} = \frac{3}{25}$.

The key idea here is independence. When two events are independent, the occurrence of one does not affect the probability of the other. This is a fundamental concept in probability, and it gives us a very clean rule for the intersection.

Think of it this way: if you flip a fair coin and roll a fair die, the coin landing heads doesn't change the chance of rolling a 4. The events are independent. The chance of both happening is just the product of their separate chances: $\frac{1}{2} \times \frac{1}{6} = \frac{1}{12}$.

The same logic applies here. We are told A and B are independent, so we don't need any complicated formulas involving conditional probability. We just multiply.

1. State the definition of independence. For any two independent events A and B, the probability that both occur is:

This is the defining property. It's not a derived formula; it's what "independent" means in mathematical terms.

2. Plug in the given values. We have $P(A) = \frac{3}{5}$ and $P(B) = \frac{1}{5}$. Substituting:

3. Multiply the fractions. Multiply the numerators together and the denominators together:

Method: Joint Probability of Independent Events

Use this whenever a problem tells you two events are independent and asks for the probability that both occur, $P(A\cap B)$.

Steps

Step 1: Recognise the independence signal

The word "independent" means the occurrence of one event does not change the probability of the other, so you may use the product form of the multiplication theorem directly — no conditional probability is needed:

Step 2: Substitute the marginal probabilities

Drop the two given individual probabilities into the product and multiply the fractions (numerator times numerator, denominator times denominator).

Step 3: Guard against the classic confusion

Do not set $P(A\cap B)=0$: that is the rule for mutually exclusive events, not independent ones. Independent events can happen together, so their joint probability is the product, generally non-zero.

Common Mistakes

Mistake 1: Confusing independent with mutually exclusive.

Students recall "mutually exclusive $\Rightarrow P(A\cap B)=0$" and answer $0$. Why it's wrong: independence is the opposite case — the events can co-occur and $P(A\cap B)=P(A)P(B)$. Correct approach: multiply, $\frac{3}{5}\times\frac{1}{5}=\frac{3}{25}$.

Mistake 2: Adding instead of multiplying.

Writing $P(A\cap B)=P(A)+P(B)$ confuses union with intersection. Why it's wrong: the intersection of independent events uses the product rule, never a sum. Correct approach: use $P(A\cap B)=P(A)\,P(B)$.