Module 3 · Statistics Foundation

Probability foundations.

This module introduces probability as the mathematical language of uncertainty. Students move from simple chance statements to event rules, conditional probability, independence and Bayesian updating. The aim is not only to calculate probabilities, but to reason clearly when information is incomplete.

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Lessons

9–10 hrs

Study time

Coding

Foundation → Advanced

Level

What this module builds

The reasoning language behind inference.

Probability is the bridge between descriptive statistics and inference. Once students understand uncertainty, events, conditions and evidence, they are ready to understand sampling distributions, confidence intervals, hypothesis tests and model uncertainty later in the course.

Uncertainty

Build probability as a formal language for uncertain outcomes, chance processes and long-run behaviour.

Events

Represent probability questions using sample spaces, outcomes, events, complements, unions and intersections.

Conditions

Understand how probability changes when information is known, using restricted sample spaces and two-way tables.

Independence

Decide whether one event changes the probability of another, and use multiplication rules carefully.

Updating

Use Bayes’ theorem to update probability when evidence appears, especially in diagnostic and risk settings.

By the end

Students should be able to reason under uncertainty.

Explain probability as a number between 0 and 1.

Define outcomes, events and sample spaces using set notation.

Use complement, union, intersection and addition rules correctly.

Calculate conditional probabilities from formulae, tables and trees.

Distinguish P(A | B) from P(B | A).

Identify independence and dependence using conditional and joint probabilities.

Apply Bayes’ theorem to diagnostic-style evidence.

Explain base-rate effects, false positives and posterior probability.

Probability reasoning workflow

Ask four questions before calculating.

Probability becomes much easier when students identify the sample space, define the event, check whether information is given and decide whether evidence should update the probability.

What is the sample space?

Every probability question begins by identifying the possible outcomes and the total reference set.

Which event is being asked about?

Translate ordinary language into event notation such as A, Aᶜ, A ∪ B or A ∩ B.

Is information already known?

If the question says 'given', the denominator changes to the condition group.

Does evidence update belief?

Bayes’ theorem combines prior probability with evidence to produce a posterior probability.

Formula map

The core rules build one system.

Students should not memorise these formulae as isolated tricks. Each rule comes from event regions, restricted sample spaces or evidence updating.

Complement

P(Aᶜ) = 1 − P(A)

Useful when it is easier to calculate what does not happen.

Addition rule

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

Subtracts the overlap so it is not counted twice.

Conditional probability

P(A | B) = P(A ∩ B) / P(B)

Calculates the probability of A inside the restricted space B.

Independence

P(A ∩ B) = P(A)P(B)

Draw the events before using the formula.

For every probability question, first sketch the sample space, label the event regions, identify the condition if there is one, and only then apply the formula. This habit prevents most common mistakes with complements, intersections, conditionals and Bayes’ theorem.

Module completion

Ready for statistical inference.

After these five lessons, students should understand uncertainty, event logic, conditional reasoning, independence and Bayesian updating. The next stage can introduce sampling distributions and inference with a stronger probability foundation.

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