# Basic probability concepts and combined events

> IB Mathematics AI SL · Unit 4: Statistics and Probability
> Source: https://www.owlsprep.com/study/ib-math-ai-sl-u4-basic-probability-concepts-and-combined/

This sub-topic introduces core fundamental probability concepts required for all IB AI SL probability problems. You will learn to work with combined events, use set notation, Venn diagrams, and apply the key addition rule for probability.

**Prerequisites:** [Basic set notation and Venn diagram interpretation](https://www.owlsprep.com/study/ib-math-ai-sl-u3-set-notation/)

## Learning objectives

- Define core probability terms including experiment, outcome, sample space and event
- Calculate probabilities for simple and combined events using set notation and Venn diagrams
- Apply the addition rule for combined events correctly
- Identify mutually exclusive events

## Key Definitions and Basic Probability Axioms

**Classical Probability** — For an experiment with equally likely outcomes, the probability of event $A$ is the ratio of the number of outcomes in $A$ to the total number of outcomes in the sample space.

*Notation:* P(A)

*Example:* Rolling a fair 6-sided die: $P(\text{roll a } 2) = \frac{1}{6}$

- Any valid probability satisfies $0 \leq P(A) \leq 1$
- The probability of the entire sample space is $P(S) = 1$, so at least one outcome must occur
- For the complement of $A$ (all outcomes not in $A$), $P(A') = 1 - P(A)$

**Worked example:** One card is drawn at random from a fair standard 52-card deck. What is the probability of drawing an ace?

1. Step 1: Identify the size of the sample space. All outcomes are equally likely.
2. $$n(S) = 52$$
3. Step 2: Count the number of outcomes in the event $A$ (drawing an ace).
4. $$n(A) = 4$$
5. Step 3: Apply the classical probability formula.
6. $$P(A) = \frac{n(A)}{n(S)} = \frac{4}{52} = \frac{1}{13}$$
7. Confirm the result is between 0 and 1, so it is valid. The probability is $\frac{1}{13}$.

## Combined Events and Set Notation

**Combined Events** — A combined event is formed from two individual events: intersection means both events occur, union means at least one event occurs.

*Notation:* Intersection: $A \cap B$, Union: $A \cup B$

*Example:* For $A = \{\text{drawing a heart}\}$ and $B = \{\text{drawing an ace}\}$, $A \cap B = \{\text{ace of hearts}\}$

Venn diagrams are the standard tool to visualize combined events in IB AI SL. The rectangle represents the full sample space, each circle represents an event, and overlapping regions represent intersections.

**Worked example:** In a class of 30 students, 18 play football, 15 play basketball, and 7 play neither sport. How many students play both sports?

1. Step 1: Calculate the number of students that play at least one sport.
2. $$n(A \cup B) = \text{Total} - n(\text{neither}) = 30 - 7 = 23$$
3. Step 2: Use the set counting formula for combined events, rearranged to solve for the intersection.
4. $$n(A \cap B) = n(A) + n(B) - n(A \cup B) = 18 + 15 - 23 = 10$$
5. Step 3: Verify the total: $8 \text{ (football only)} + 10 \text{ (both)} + 5 \text{ (basketball only)} + 7 \text{ (neither)} = 30$, which matches. 10 students play both sports.

## Addition Rule and Mutually Exclusive Events

> **info**
>
> The addition rule for probability is one of the most frequently tested concepts in IB AI SL probability questions, so memorize it correctly.

**Mutually Exclusive Events** — Two events are mutually exclusive if they cannot occur at the same time, so their intersection is empty and $P(A \cap B) = 0$.

*Notation:* $A \cap B = \emptyset$

*Example:* Rolling a die: events 'rolling a 2' and 'rolling an odd number' are mutually exclusive.

The general addition rule works for any two events, whether mutually exclusive or not. Dividing the set counting rule by the total sample size gives the probability form:

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

For mutually exclusive events, the rule simplifies to $P(A \cup B) = P(A) + P(B)$ because $P(A \cap B) = 0$.

**Worked example:** A 12-section spinner numbered 1 to 12 is spun once. Event A = spinning a multiple of 3, Event B = spinning a multiple of 4. Are A and B mutually exclusive? Find $P(A \cup B)$.

1. Step 1: List outcomes for each event: $A = \{3, 6, 9, 12\}$, $B = \{4, 8, 12\}$
2. Step 2: Check the intersection: $A \cap B = \{12\}$, so $P(A \cap B) = \frac{1}{12} \neq 0$. The events are not mutually exclusive.
3. Step 3: Calculate individual probabilities:
4. $$P(A) = \frac{4}{12}, \quad P(B) = \frac{3}{12}$$
5. Step 4: Apply the general addition rule:
6. $$P(A \cup B) = \frac{4}{12} + \frac{3}{12} - \frac{1}{12} = \frac{6}{12} = \frac{1}{2}$$

## Common pitfalls

- **Wrong:** Forgetting to subtract $P(A \cap B)$ when applying the addition rule
  - Why it fails: This double-counts the overlapping region, leading to a probability that is too high, sometimes even greater than 1 which is impossible
  - Correct: Always include the $-P(A \cap B)$ term unless you have confirmed events are mutually exclusive
- **Wrong:** Assuming all events are mutually exclusive by default
  - Why it fails: Most combined events have overlapping outcomes, so simplifying the addition rule by default leads to incorrect results
  - Correct: Always explicitly check for common outcomes before assuming events are mutually exclusive
- **Wrong:** Confusing intersection ($A \cap B$) and union ($A \cup B$) notation
  - Why it fails: Mixing these up changes the entire question request, leading to wrong answers even if calculations are correct
  - Correct: Remember: Intersection = 'And' (both events occur, looks like an 'n')
- **Wrong:** Forgetting to include outcomes outside both events when checking total probability
  - Why it fails: Missing the outer region of a Venn diagram leads to incorrect counts and probabilities
  - Correct: Always confirm that the sum of all probabilities in your Venn diagram equals 1 before calculating your final answer

## Cheatsheet

| Concept | Notation | Formula/Rule |
| --- | --- | --- |
| Complement of $A$ | $A'$ | $P(A') = 1 - P(A)$ |
| Intersection ($A$ and $B$) | $A \cap B$ | $P(A \cap B)$ = probability both occur |
| Union ($A$ or $B$, at least one) | $A \cup B$ | $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ |
| Mutually exclusive events | $A \cap B = \emptyset$ | $P(A \cap B) = 0$, $P(A \cup B) = P(A)+P(B)$ |
| Classical Probability | $P(A)$ | $P(A) = \frac{n(A)}{n(S)}$ |

## What's next

Mastering basic probability concepts and combined events is the foundation for all further probability topics in IB AI SL. Every probability problem you encounter will build on the notation, rules and Venn diagram skills you learned here. Next, you will extend these concepts to conditional probability, which accounts for extra information about events, then use these skills to build tree diagrams for sequential events, and finally move on to probability distributions for random variables. Strong fluency with this sub-topic will make all future probability work much easier.

- [Discrete Probability Distributions](https://www.owlsprep.com/study/ib-math-ai-sl-u4-discrete-probability-distributions/)
- [Conditional probability and independence](https://www.owlsprep.com/study/ib-math-ai-sl-u4-conditional-probability-and-independence/)
- [Binomial distribution](https://www.owlsprep.com/study/ib-math-ai-sl-u4-binomial-distribution/)

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