Independence

Distinguish true independence from mere lack of correlation, and verify it with the multiplication rule.

1. The idea

Events $A$ and $B$ are independent if knowing one occurred gives no information about the other: $P (A \cap B) = P (A) P (B)$ , equivalently $P (A ∣ B) = P (A)$ whenever $P (B) > 0$ .

Independence is a strong, structural property — it is not the same as being uncorrelated, which only concerns linear association between random variables. Two variables can be uncorrelated yet highly dependent (e.g. $Y = X^{2}$ for symmetric $X$ ), while independent variables are always uncorrelated.

The most reliable way to check independence in an interview is to compute all three probabilities — $P (A)$ , $P (B)$ , $P (A \cap B)$ — directly from the sample space rather than relying on intuition, since many pairs of events that seem related turn out to be exactly independent, and vice versa.

2. Why it matters

Independence claims are a classic interview trap: intuition often misleads, and interviewers want to see the multiplication-rule check performed explicitly rather than assumed.

3. Core mathematics

Independence definition

P (A \cap B) = P (A) P (B)

Equivalent form

P (A ∣ B) = P (A)

Pairwise vs mutual independence

P (A \cap B \cap C) = P (A) P (B) P (C)

Mutual independence requires this for every subset, not just pairwise products.

4. How to recognize it

Think of this when…

The problem asks you to verify or disprove that two events are independent.
A claim of independence is used to justify multiplying probabilities — check it first.
Uncorrelated random variables are described, but the question is about independence, not correlation.
Three or more events are given and pairwise independence is checked but mutual independence is in question.

5. Worked example

Worked example

Roll a fair die. Let $A$ = "the die shows an even number" and $B$ = "the sum with a second independent fair die roll is even." Are $A$ and $B$ independent?

1
Compute $P (A) = 3/6 = 1/2$ (die shows 2, 4, or 6).
2
The sum of two dice is even iff both dice are even or both are odd, so $P (B) = P (both even) + P (both odd) = \frac{1}{4} + \frac{1}{4} = \frac{1}{2}$ .
3
Compute $P (A \cap B)$ : first die even and sum even means second die must also be even, probability $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$ .
$P (A \cap B) = \frac{1}{4}$
4
Compare to $P (A) P (B) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$ ; they match, so $A$ and $B$ are independent.
$P (A \cap B) = P (A) P (B)$
Answer: Yes — $A$ and $B$ are independent, since $P (A \cap B) = 1/4 = P (A) P (B)$ .

6. Common mistakes

Assuming two events are dependent just because they both involve the same die or variable — always check the multiplication rule.
Treating uncorrelated variables as independent; correlation only measures linear dependence.
Assuming pairwise independence among three events implies full mutual independence.
Confusing independence with mutual exclusivity — mutually exclusive events with positive probability are always dependent.

7. Quick check

If $X$ is symmetric around 0 (e.g. standard normal) and $Y = X^{2}$ , are $X$ and $Y$ independent?

Events A and B are mutually exclusive with $P (A), P (B) > 0$ . Are they independent?

8. Practice

16 QuantSolve questions use this concept.

Practice this concept

9. Related methods

Complement Rule

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