Q.Let E and F be events with $P(E) = \frac{3}{5}$, $P(F) = \frac{3}{10}$ and $P(E \cap F) = \frac{1}{5}$. Are E and F independent?

Concept: Event Independence — Two events are independent if and only if $P(E \cap F) = P(E) \cdot P(F)$.

Step 1: Compute the product of the individual probabilities:

Step 2: Compare this with the given intersection probability:

$P(E \cap F) = \frac{1}{5} = \frac{10}{50}.$ …

The idea of independence is simple: two events are independent if knowing that one has occurred gives you no information about whether the other has occurred. In probability terms, this means the probability that both happen together is exactly the product of their individual probabilities. That’s the test we’ll use.

Let’s check it step by step.

1. Write down what’s given. We have

2. Compute the product $P(E) \cdot P(F)$. Multiply the two probabilities:

3. Compare with $P(E \cap F)$. The given intersection probability is $\frac{1}{5}$. Write it with denominator 50 to compare easily:

So we have $P(E \cap F) = \frac{10}{50}$ and $P(E) \cdot P(F) = \frac{9}{50}$.

4. Apply the independence condition. For independence, we need $P(E \cap F) = P(E) \cdot P(F)$. Here $\frac{10}{50} \neq \frac{9}{50}$, so the condition fails. …

Method: Independence Test from Given Probabilities

Use this when $P(A)$, $P(B)$ and $P(A\cap B)$ are all supplied numerically and you only need to declare whether the events are independent.

Steps

Step 1: Form the product of the marginals

Multiply the two individual probabilities to get $P(A)\,P(B)$ — the value the intersection would have if the events were independent.

Step 2: Put both quantities over a common denominator …

Common Mistakes

Mistake 1: Comparing fractions without a common denominator.

Students eyeball $\frac{1}{5}$ against $\frac{9}{50}$ and call them equal. Why it's wrong: they are not — $\frac{1}{5}=\frac{10}{50}\neq\frac{9}{50}$. Correct approach: convert to a common denominator before deciding, giving not independent.

Mistake 2: Thinking $P(E\cap F)\neq0$ proves independence. …