Understanding Statistical Independence

Concept

Statistical independence is a key principle in probability theory. It describes a scenario where two events are independent if the occurrence of one event does not affect the probability of the other event happening.

For two events A and B, they are independent if the probability of both events occurring simultaneously is equal to the product of their individual probabilities:

Formal Definition

Two events A and B are considered independent if and only if:

P(A ∩ B) = P(A) ⋅ P(B)

Here:

• P(A ∩ B): Probability that both events A and B occur simultaneously.
• P(A): Probability that event A occurs.
• P(B): Probability that event B occurs.

Example: Rolling Two Dice

Consider rolling two fair dice. Let:

• A: The event that the first die shows a 4.
• B: The event that the second die shows a 6.

Since the outcome of one die does not affect the outcome of the other, events A and B are independent. Their probabilities are calculated as follows:

• P(A) = 1/6 (probability that the first die shows 4).
• P(B) = 1/6 (probability that the second die shows 6).
• P(A ∩ B) = 1/6 × 1/6 = 1/36 (probability that both dice show the specified values).

Since P(A ∩ B) = P(A) ⋅ P(B), events A and B are independent.

Analogy with Conditional Probability

If two events A and B are independent, then knowing that B has occurred does not change the probability of A occurring:

P(A | B) = P(A) and similarly P(B | A) = P(B).

This means that for independent events, the occurrence of one event provides no additional information about the likelihood of the other event occurring.