• Describe and find conditional probabilities.
• Understand the concept of independent events.

Conditional Probabilities

A conditional probability is calculated based on the assumption that one event has already occurred. Conditional probabilities restrict the total. The new total is indicated after the word GIVEN in the question.

Sometimes, when the probability problems are complex, it can be helpful to present the situation in a contingency table. Additionally, some people prefer to use formulas in finding probabilities.

Formulas:

• Conditional probability: For any events [latex]A[/latex] and [latex]B[/latex] where [latex]P(B)>0[/latex],

[latex]P(A \text{ GIVEN } B) = \frac{P(A \text{ and } B)}{P(B)}[/latex]

• Independent events: Two events are independent if [latex]P(A \text{ GIVEN } B) = P(A)[/latex].

Alternatively, two events are independent if:

[latex]P(A \text{ GIVEN } B) = \frac{P(A \text{ and } B)}{P(B)}[/latex]

[latex]P(A) = \frac{P(A \text{ and } B)}{P(B)}[/latex]

Multiply both sides with [latex]P(B)[/latex]:

[latex]P(A) \times P(B) = P(A \text{ and } B)[/latex]