IB Mathematics: AA HL topic guide

Statistics and Probability

Statistics and Probability is a core part of IB Mathematics: AA HL. This guide connects the syllabus ideas behind Introduction to Statistics and Probability, Sampling and Presenting Data, Measures of Central Tendency and Dispersion, Correlation and Regression, Probability Fundamentals and 5 more units, shows how they appear in worked problems, and points you to the formulas and full lessons needed for exam revision.

What you will learn

Statistics and Probability syllabus outline

The units below follow the structure used in the full Study to Learn course. Use the outline to identify exactly which idea needs attention, then work through the public example before continuing to the complete lesson path.

4.0

Introduction to Statistics and Probability

What You Will Learn in Statistics and Probability

4.1

Sampling and Presenting Data

Sampling Techniques · Presenting Data

4.2

Measures of Central Tendency and Dispersion

Mean, Median & Mode · Range, IQR, Variance & Standard Deviation

4.3

Correlation and Regression

Scatter Diagrams & Pearson's r · The Regression Line

4.4

Probability Fundamentals

Sample Space & Basic Probability · Venn/Tree Diagrams & Conditional Probability

4.5a

Discrete Random Variables and Expected Value

Discrete Random Variables & Expected Value

4.5b

The Binomial Distribution

The Binomial Distribution · Binomial Distribution — Applications and Calculations

4.6

The Normal Distribution

Standardization & Probabilities · Inverse Normal and Applied Problems

4.7

Bayes' Theorem

Bayes' Theorem · Bayes' Theorem with Tree Diagrams

4.8

Further Expectation and Variance

Linear Combinations of Random Variables · Applications to Sums, Differences and Real-World Contexts

Free worked preview

Bayes' Theorem

This complete preview comes from the Bayes' Theorem unit. It introduces the core language, shows the method in context, and gives you a real example of the lesson quality before you create an account.

Bayes' Theorem

Bayes' theorem answers a question that is surprisingly subtle: given that you observed some evidence B, what is the revised probability that hypothesis A is true? This "inversion" of conditional probability — finding P(A|B) from P(B|A) — is the mathematical engine behind medical diagnosis (given a positive test, what is the probability of actually having the disease?), spam filtering, and machine learning. The counterintuitive results (a "95% accurate" test can still mean a low probability of actually having a rare disease) make Bayes' theorem one of the most practically important — and most frequently misunderstood — results in the syllabus. By the end you will be able to apply Bayes' theorem to reverse conditional probabilities, use the law of total probability to find P(B), and interpret the often-surprising results in context.

Bayes' theorem lets you "reverse" a conditional probability — finding P(A|B) when you actually know P(B|A) instead.

Bayes' Theorem (Two Events)
P(A∣B)=P(B∣A)P(A)P(B)P(A|B) = \frac{P(B|A)P(A)}{P(B)}

Often P(B) itself must be expanded using the law of total probability:

P(B)=P(B∣A)P(A)+P(B∣A′)P(A′)P(B) = P(B|A)P(A) + P(B|A')P(A')
Tree Diagram Approach

Bayes' theorem problems are usually easiest with a tree diagram: draw the "given" probabilities first (e.g. P(A) then P(B|A)), multiply along branches to get P(A∩B), then Bayes' theorem is just "the branch you want" divided by "the sum of all branches ending in B".

Common ErrorP(A|B) and P(B|A) are generally very different quantities — this confusion (sometimes called "the prosecutor's fallacy" in real-world contexts) is exactly what Bayes' theorem exists to correctly resolve.
Worked Example A medical test is 95% accurate for people with a disease (true positive rate) and has a 3% false positive rate for people without it. 1% of the population has the disease. Given a positive test, find the probability the person actually has the disease.
1Set up known probabilitiesP(D)=0.01, P(+∣D)=0.95, P(+∣D′)=0.03P(D)=0.01,\ P(+|D)=0.95,\ P(+|D\prime)=0.03
2Law of total probability for P(+)P(+)=0.95(0.01)+0.03(0.99)=0.0095+0.0297=0.0392P(+) = 0.95(0.01)+0.03(0.99) = 0.0095+0.0297=0.0392
3Apply Bayes' theoremP(D∣+)=0.00950.0392P(D|+) = \frac{0.0095}{0.0392}
P(D∣+)≈0.242 — only about 24%, despite the test seeming "95% accurate"P(D|+) \approx 0.242\text{ — only about 24\%, despite the test seeming "95\% accurate"}
Practice QuestionBox A has 3 red, 2 blue balls; Box B has 1 red, 4 blue. A box is chosen at random and a red ball is drawn. Find the probability the ball came from Box A.

Use the formulas

Keep the formula booklet beside you

Revision becomes faster when you practise locating the right equation and understand what every variable means. The full reference is public and searchable on any device.

Open IB Math AA HL Formula Booklet →

Reviewed by the Study to Learn editorial team · Updated 2026-07-24