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Independent Events

     - The Simplest Case of Conditional Probability
     - The Simplest Case of the Multiplication Rule

Two events A and B are said to be independent if the occurence of one does not affect the occurence of the other. In other words, the probability that A occurs remains the same whether B has occurred or not. Moreover, the same is true in reverse. The probability that B occurs remains unaffected by the occurrence of A. The happenings of independent events are completely independent of one another.

The conditional probability of independent events simplifies to their original probability of happening.

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

The Multiplication Rule for independent events simplifies to:

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

NOTE:  This does not mean that independent events cannot happen at the same time. In fact, independent events MUST be able to occur concurrently. Otherwise, if A and B are independent and mutually exclusive, then, by the Multiplication Rule,

P(A and B) = P(A)· P(B) = 0 .

This, in turn, implies that either P(A) = 0 or P(B) = 0 or both are 0, meaning that one or the other or both of the events cannot occur at all.

Therefore, events CANNOT be both mutally exclusive and independent!

Example

A local supermarket has two computers on its premises which are not networked. One is newer than the other. The probability that the newer one will break down in any given month is 0.04. The probability that the older one will break down in any given month is 0.07.

  1. Are these events independent?
  2. Are these events mutually exclusive?
  3. What is the probability that the two computers will both break down in any given month?
  4. What is the probability that either one or the other will break down in any given month?

Solution

Let A be the event that the newer computer breaks down in any given month.
Let B be the event that the older computer breaks down in any given month.
Then P(A) = 0.04  and  P(B) = 0.07.

Answer to 1:

A and B are independent events since either one of the computers is not affected by the breakdown of the other computer. They operate independently.

Answer to 2:

A and B are not mutually exclusive since there is a possibility that both computers could break down in any given month. Moreover, independent events cannot be mutually exclusive (see the argument above).

Answer to 3:

The probability that both computers will break down in any given month is obtained using the Multiplication Rule for independent events.

P(A and B) = P(A) · P(B) = 0.04 · 0.07 = 0.0028.

Answer to 4:

The probability that either one or the other will break down in any given month is obtained using the Addition Rule.

P(A or B) = P(A) + P(B) - P(A and B) = 0.04 + 0.07 - 0.0028 = 0.1072 .

 

Last Updated by SAS October 07, 2026, 16:00:54