# Basic probability concepts and rules

> IB Mathematics: Analysis and Approaches HL · Statistics & Probability
> Source: https://www.owlsprep.com/study/ib-math-aa-hl-u4-basic-probability-concepts-and-rules/

This module introduces core probability terminology, key calculation rules for combined events, and core visual tools like Venn diagrams. It forms the essential foundation for all advanced probability topics tested in IB AA HL.

**Prerequisites:** [Basic set theory operations](https://www.owlsprep.com/study/ib-math-aa-hl-u3-set-theory-basics/)

## Learning objectives

- Define key probability terms including sample space, events and complementary events
- Distinguish between mutually exclusive and independent events
- Apply the complement, addition and multiplication rules correctly
- Use Venn diagrams to visualize and solve combined probability problems

## Core Probability Terminology

**Sample Space** — The set of all possible distinct outcomes of a random experiment. All outcomes must be mutually exclusive and cover every possible result.

*Notation:* S

*Example:* For rolling a single 6-sided die, $S = \{1, 2, 3, 4, 5, 6\}$

**Event** — An event is any subset of the sample space. We are usually interested in the probability of a specific event occurring.

*Notation:* A, B, ...

*Example:* For rolling a die, the event $E = \text{rolling an even number} = \{2,4,6\}$

**Worked example:** A random experiment consists of tossing two fair coins. Write down the sample space, and identify the event $A = \text{at least one coin lands on heads}$.

1. Each coin can land on heads ($H$) or tails ($T$), so we list all ordered outcomes:
2. $$S = \{(H,H), (H,T), (T,H), (T,T)\}$$
3. Event $A$ includes all outcomes with at least one head, so we select all outcomes that match the condition:
4. $$A = \{(H,H), (H,T), (T,H)\}$$

## Complementary Events and the Complement Rule

**Complementary Event** — The complement of event $A$ is the event that $A$ does **not** occur. $A'$ contains all outcomes in the sample space that are not in $A$.

*Notation:* A'

*Example:* If $A = \text{at least one 6 in two dice rolls}$, $A' = \text{no 6s in two dice rolls}$

The complement rule follows directly from the fact that $P(S) = 1$:

$$P(A) + P(A') = 1 \implies P(A) = 1 - P(A')$$

This rule is especially useful for calculating "at least one" probabilities, where calculating the complement is much simpler than calculating the event directly.

**Worked example:** Calculate the probability of getting at least one 6 when rolling two fair 6-sided dice.

1. Let $A = \text{at least one 6}$. The complement is $A' = \text{no 6s on either die}$.
2. There are $6 \times 6 = 36$ total possible outcomes. For no 6s, each die has 5 possible outcomes:
3. $$\text{Number of outcomes in } A' = 5 \times 5 = 25$$
4. Calculate $P(A')$, then apply the complement rule:
5. $$P(A') = \frac{25}{36} \implies P(A) = 1 - \frac{25}{36} = \frac{11}{36}$$

## Addition Rule for Combined Events

**Mutually Exclusive (Disjoint) Events** — Two events are mutually exclusive if they cannot occur at the same time, meaning they have no overlapping outcomes. For mutually exclusive events, $A \cap B = \emptyset$ so $P(A \cap B) = 0$.

The general addition rule calculates the probability that either $A$ or $B$ (or both) occur:

$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$

If $A$ and $B$ are mutually exclusive, this simplifies to $P(A \cup B) = P(A) + P(B)$, since $P(A \cap B) = 0$.

**Worked example:** In a class of 30 students, 16 study Mathematics, 12 study Physics, and 8 study both subjects. What is the probability a randomly selected student studies Mathematics or Physics?

1. Let $M = \text{student studies Math}$, $P = \text{student studies Physics}$.
2. From the problem, we have:
3. $$P(M) = \frac{16}{30}, \quad P(P) = \frac{12}{30}, \quad P(M \cap P) = \frac{8}{30}$$
4. Events are not mutually exclusive (some students study both), so use the general addition rule:
5. $$P(M \cup P) = \frac{16}{30} + \frac{12}{30} - \frac{8}{30} = \frac{20}{30} = \frac{2}{3}$$

## Independent Events and the Multiplication Rule

**Independent Events** — Two events are independent if the occurrence of one does not change the probability of the other occurring. The formal condition for independence is: $P(A \cap B) = P(A)P(B)$.

For independent events, the multiplication rule gives the probability that both events occur:

$$P(A \cap B) = P(A) \times P(B)$$

**Worked example:** A bag contains 5 red marbles and 3 blue marbles. One marble is drawn at random, replaced, then a second marble is drawn. What is the probability both marbles are red?

1. Let $R_1 = \text{first marble red}$, $R_2 = \text{second marble red}$.
2. Since the first marble is replaced, the outcome of the first draw does not affect the second, so events are independent.
3. $$P(R_1) = P(R_2) = \frac{5}{8}$$
4. Apply the multiplication rule for independent events:
5. $$P(R_1 \cap R_2) = P(R_1) \times P(R_2) = \frac{5}{8} \times \frac{5}{8} = \frac{25}{64}$$

> **note**
>
> If the first marble was **not replaced**, the events would be dependent, and this simple multiplication rule would not apply. Dependent events are covered in conditional probability.

## Common pitfalls

- **Wrong:** Confusing mutually exclusive events and independent events
  - Why it fails: Mutually exclusive describes overlap of outcomes, while independent describes a probability relationship between events. These are unrelated concepts.
  - Correct: Remember: If two events have non-zero probability and are mutually exclusive, they cannot be independent. Always check the definition for each term separately.
- **Wrong:** Forgetting to subtract $P(A \cap B)$ when using the addition rule
  - Why it fails: If events are not mutually exclusive, overlapping outcomes are counted twice when you add $P(A)$ and $P(B)$.
  - Correct: Always use the general addition rule $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ unless explicitly told the events are disjoint.
- **Wrong:** Applying the simple multiplication rule to dependent events
  - Why it fails: The rule $P(A \cap B) = P(A)P(B)$ only holds for independent events.
  - Correct: Only use the simple multiplication rule when events are confirmed to be independent (e.g. sampling with replacement, independent trials).
- **Wrong:** Assuming all outcomes in a sample space are equally likely
  - Why it fails: Many random experiments do not have uniform probability for outcomes, so counting outcomes directly will give the wrong result.
  - Correct: Only assume equally likely outcomes if the problem explicitly states the experiment is fair or selection is uniform.

## Cheatsheet

| Rule/Concept | Formula | When To Use |
| --- | --- | --- |
| Complement Rule | $P(A) = 1 - P(A')$ | Calculating "at least one" probabilities |
| General Addition Rule | $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ | Any two events |
| Addition Rule (Mutually Exclusive) | $P(A \cup B) = P(A) + P(B)$ | Disjoint events with no overlap |
| Multiplication Rule (Independent) | $P(A \cap B) = P(A)P(B)$ | Independent events |
| Independence Check | $P(A \cap B) = P(A)P(B)$ | Confirming if two events are independent |

## What's next

Now that you have mastered basic probability concepts and rules, you have the essential foundation to explore all advanced probability topics tested in IB AA HL. The rules you learned here will be extended to conditional probability, where we calculate probabilities for events given that another event has already occurred, and to multi-step probability problems using tree diagrams and Bayes' theorem. You will also apply these core rules to discrete and continuous probability distributions later in the unit, which are heavily weighted topics in both Paper 1 and Paper 2. Mastery of these basic concepts is critical to avoid losing easy marks, as basic probability often forms the first part of longer multi-part exam questions.

- [Discrete Probability Distributions](https://www.owlsprep.com/study/ib-math-aa-hl-u4-discrete-probability-distributions/)
- [Conditional probability and Bayes' theorem](https://www.owlsprep.com/study/ib-math-aa-hl-u4-conditional-probability-and-bayes-theorem/)
- [Binomial distribution](https://www.owlsprep.com/study/ib-math-aa-hl-u4-binomial-distribution/)

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