Basic probability concepts and combined events

IB Mathematics AI SL· IB AI SL 4.6· 15 min read

1. Key Definitions and Basic Probability Axioms★☆☆☆☆⏱ 15 min

📘 Definition

Classical Probability

P(A)

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.

Example:

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

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

2. Combined Events and Set Notation★★☆☆☆⏱ 20 min

📘 Definition

Combined Events

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

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

Example:

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

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

3. Addition Rule and Mutually Exclusive Events★★☆☆☆⏱ 20 min

📘 Definition

Mutually Exclusive Events

$A \cap B = \emptyset$

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

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 $P (A \cup B)$ .

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Step 1: List outcomes for each event: $A = {3, 6, 9, 12}$ , $B = {4, 8, 12}$
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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.
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Step 3: Calculate individual probabilities:
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$P(A) = \frac{4}{12}, \quad P(B) = \frac{3}{12}$
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Step 4: Apply the general addition rule:
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$P(A \cup B) = \frac{4}{12} + \frac{3}{12} - \frac{1}{12} = \frac{6}{12} = \frac{1}{2}$

4. Common Pitfalls

Wrong move:

Forgetting to subtract $P (A \cap B)$ when applying the addition rule

Why:

This double-counts the overlapping region, leading to a probability that is too high, sometimes even greater than 1 which is impossible

Correct move:

Always include the $- P (A \cap B)$ term unless you have confirmed events are mutually exclusive

Wrong move:

Assuming all events are mutually exclusive by default

Why:

Most combined events have overlapping outcomes, so simplifying the addition rule by default leads to incorrect results

Correct move:

Always explicitly check for common outcomes before assuming events are mutually exclusive

Wrong move:

Confusing intersection ( $A \cap B$ ) and union ( $A \cup B$ ) notation

Why:

Mixing these up changes the entire question request, leading to wrong answers even if calculations are correct

Correct move:

Remember: Intersection = 'And' (both events occur, looks like an 'n')

Wrong move:

Forgetting to include outcomes outside both events when checking total probability

Why:

Missing the outer region of a Venn diagram leads to incorrect counts and probabilities

Correct move:

Always confirm that the sum of all probabilities in your Venn diagram equals 1 before calculating your final answer

5. Quick Reference 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	$P (A)$	$P (A) = \frac{n ( A )}{n ( S )}$

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2025 · 1
Combined event probability calculation
2023 · 2
Venn diagram probability problem

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.