# Probability of Two or More Events

• The addition rule applies when considering the probability of either of two events happening.
• It states the probability of either event A or event B occurring is equal to the sum of their individual probabilities, minus the probability of them both occurring.
• P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

## Multiplication Rule

• The multiplication rule applies when considering the probability of two events both occurring.
• It states that the probability of both event A and event B happening is the product of the probability of A happening and the probability of B happening given A has happened.
•  **P(A ∩ B) = P(A) × P(B A)**

## Independent Events

• Two events are independent if the outcome of one event does not affect the outcome of the other.
• For independent events, the multiplication rule simplifies to:
• P(A ∩ B) = P(A) × P(B)

## Dependent Events

• Two events are dependent if the outcome of one event affects the outcome of the other.
•  If A and B are dependent events, the probability of both occurring is given by **P(A ∩ B) = P(A) × P(B A)**, where P(B A) is the conditional probability of B given A has happened.

## Mutually Exclusive Events

• Two events are mutually exclusive if they cannot both occur at the same time.
• For mutually exclusive events, P(A ∩ B) = 0.

## Using Venn Diagrams

• Venn diagrams can be a helpful tool for modelling the overlap between events and visualising probabilities.
• Each circle represents an event, and their overlap (if present) represents the joint occurrence of these events.
• Probabilities should be written in the appropriate sections of the diagram, with the entire ‘universe’ (all possible outcomes) summing to 1.

## Using Contingency Tables

• A contingency table or cross-tabulation is a type of table used in statistics to summarise categorical data.
• It allows you to quickly calculate the probabilities of combined and conditional events.
• Each row represents an outcome of one event and each column represents an outcome of the other event. The entries in the cells of the table are the joint probabilities of the corresponding outcomes.

Remember, solving probability problems often involves combining these principles, so practice using different combinations of them.