Independent Events

Independent events are events in probability in which the happening of one event does not affect the happening of the other event. This concept is important because many board questions ask students to identify whether events are independent and then apply the correct probability formula.

Two events are said to be independent if the occurrence of one does not depend on the other.

For two events E and F:

• E and F are independent if P(F | E) = P(F), when \[P(E) \neq 0\].

• Similarly, E and F are independent if P(E | F) = P(E), when \[P(F) \neq 0\].

• An equivalent and most commonly used test is:

• If E and F are independent, then E and F' are also independent.

• If E and F are independent, then E' and F are also independent.

• If E and F are independent, then E' and F' are also independent.

• For independent events A and B, the probability that at least one occurs is:

Die Thrown Once

A die is thrown once. Let:

• E: the number appearing is a multiple of 3

• F: the number appearing is even

Step 1: Write the sample space

Step 2: Write each event as a set

• E = {3, 6}, because 3 and 6 are multiples of 3.

• F = {2, 4, 6}, because 2, 4, and 6 are even numbers.

Step 3: Find the common outcomes

• Independent events do not influence each other.

• The standard test is \[P(E \cap F) = P(E)P(F)\].

• Conditional form: \[P(F | E) = P(F)\] and \[P(E | F) = P(E)\], when defined.

• If two events are independent, related complement pairs are also independent.

• Mutually exclusive events and independent events are different.

• For independent events A and B, \[P(A \cup B) = 1 - P(A')P(B')\].

• For three events, mutual independence requires pairwise conditions and the condition involving all three together.