The Addition Law of Probability

Example: The Student Scenario Imagine a class of 30 students. There are 17 girls and 5 students who are "A students." However, 3 of those A students are also girls. If we want to find the probability of picking a student who is either a girl or an A student, we calculate: $P(\text{Girl or A student}) = P(\text{Girl}) + P(\text{A student}) - P(\text{Girl and A student})$ $P(\text{Girl or A student}) = \frac{17}{30} + \frac{5}{30} - \frac{3}{30} = \frac{19}{30} \approx 0.6333$

Mutually Exclusive Events

A vital distinction to make when using the addition law is whether the events are mutually exclusive. Two events are considered mutually exclusive if they have no outcomes in common; they cannot happen at the same time. Mathematically, for mutually exclusive events, the intersection is a null event, meaning: $P(B \text{ and } C) = 0$

When events are mutually exclusive, the general addition law simplifies into a special case: $P(B \text{ or } C) = P(B) + P(C)$

For instance, if you are selecting a person from a group, the events "Male" and "Female" are mutually exclusive because an individual cannot belong to both categories simultaneously in a given dataset. Conversely, being "in favor of a policy" and "being a specific gender" are not mutually exclusive, as a person can be both a female and in favor of the policy.

Using Contingency Tables

In real-world data analysis, probabilities are often organized into contingency tables. These tables allow us to systematically view the frequencies of different categorical variables, making it easier to identify both marginal and joint probabilities.

Consider a study of 100 people regarding their opinion on the death penalty, categorized by gender:

From this table, we can derive several types of probabilities:

1. Marginal Probability: This is the probability of a single event without considering any other variables (e.g., the probability of being Male is $60/100$).
2. Joint Probability: This is the probability of two events occurring together (e.g., the probability of being Male and in favour is $15/100$).
3. Conditional Probability: This is the probability of an event occurring given that another event has already happened (e.g., the probability of being in favour given the person is Male).

Exam Tip: Identifying the Law When approaching a probability question, always identify the "operative word" first:

• If the question asks for "OR", focus on the Addition Law.
• If the question asks for "AND", focus on the Multiplication Law.

By understanding whether events overlap and how they are structured within a sample space, you can apply the Addition Law accurately to solve complex probability problems.