Q.A die is thrown once. The number on the die is a multiple of $3$ is denoted by $E$, and the number on the die is even is denoted by $F$. Are $E$ and $F$ independent events?

Event Independence

Two events are independent when the occurrence of one does not change the probability of the other. Toss a coin and roll a die: the coin landing heads tells you nothing about whether the die shows a six. Contrast this with drawing cards without replacement, where the first draw does change the odds for the second — those events are dependent.

From Conditional Probability to a Clean Test

"Knowing $B$ doesn't change $A$" means $P(A \mid B) = P(A)$. Substituting the definition $P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}$ and clearing the fraction gives the symmetric form used in practice:

Events $A$ and $B$ are independent exactly when the probability of both occurring equals the product of their individual probabilities. This version is preferred because it needs no non-zero condition and treats $A$ and $B$ alike.

A Quick Check

Roll a fair die. Let $A = \{2,4,6\}$ (even) and $B = \{4,5,6\}$ (greater than 3). Then $P(A) = P(B) = \tfrac{1}{2}$, and $A \cap B = \{4,6\}$ so $P(A \cap B) = \tfrac{1}{3}$. Since $\tfrac{1}{3} \neq \tfrac{1}{2} \cdot \tfrac{1}{2} = \tfrac{1}{4}$, these events are not independent.

Three or More Events

Events $A, B, C$ are mutually independent only if all four conditions hold: the three pairwise products and

Pairwise independence alone is not enough to guarantee mutual independence. …