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Mathematics · 6. Probability and statistics

From randomness to models and evidence

Build sample spaces, distinguish independence from mutual exclusivity and use simulation to understand sampling variation.

40 minutesContent version: 2.1
Start the interactive pathwayOpen the virtual experiment

Driving question

When is simulation close enough to theory, and how much deviation is normal?

Curriculum coverage

Vietnam Mathematics 6–12; Cambridge IGCSE/A-Level probability and statistics.

Model scope and limitations

Pseudorandom simulation illustrates a model; one finite run does not prove a distribution or replace a statistical test.

Learning objectives

  • Construct a non-overlapping sample space.
  • Calculate union, intersection and conditional probability.
  • Explain why relative frequency converges while still fluctuating.

Prerequisite knowledge

  • Fractions, basic counting and reading tables/graphs.

Core knowledge

Independence

A and B are independent when P(A∩B)=P(A)P(B), equivalently P(A|B)=P(A) when P(B)>0.

Mutual exclusivity

Mutually exclusive events cannot occur together; if both have positive probability, they are not independent.

Law of large numbers

As trials increase, relative frequency tends toward probability but need not approach monotonically.

Worked example

For two fair coin tosses, Ω={HH,HT,TH,TT}. Exactly one head has probability 2/4 = 1/2; order prevents missing HT or TH.

Misconceptions to avoid

  • After five tails, a fair coin still has P(head)=1/2; the past is not “owed” a correction.
  • Correlation does not prove causation; confounders can create association.

Virtual experiment procedure: Galton board and binomial distribution

  1. State the model, fairness/independence assumptions and predicted distribution before running.
  2. Run increasing sample sizes and save frequencies, not only the final picture.
  3. Compare frequency with probability using absolute deviation and explain variation.

Safety and cautions

  • Do not interpret small probability as impossible, or use an educational simulation for medical/financial decisions.

Evidence to collect

  • A sample-size/frequency table, distribution graph and comment on both trend and fluctuation.
Open the virtual experiment

Diagnostic check

1. For mutually exclusive A and B, P(A∩B)=?

  1. 0
  2. 1
  3. P(A)+P(B)
Show answer and explanation

Correct answer: 0

They cannot occur together.

2. Conditional probability P(A|B) equals?

  1. P(A∩B)/P(B)
  2. P(A)+P(B)
  3. P(B)/P(A∩B)
Show answer and explanation

Correct answer: P(A∩B)/P(B)

Restricting the sample space to B gives division by P(B).

Application practice

1. For a fair die, P(even)=?

  1. 1/2
  2. 1/6
  3. 2/3
Show answer and explanation

Correct answer: 1/2

There are three even outcomes among six equally likely outcomes.

2. If independent P(A)=0.4 and P(B)=0.5, P(A∩B)=?

  1. 0.2
  2. 0.9
  3. 0.45
Show answer and explanation

Correct answer: 0.2

For independence multiply: 0.4×0.5 = 0.2.

3. What does increasing simulation trials usually do?

  1. Reduce relative variation
  2. Guarantee exact agreement
  3. Change true probability
Show answer and explanation

Correct answer: Reduce relative variation

Proportions usually stabilise but still show random fluctuation.

Academic references

  1. OpenStax Precalculus 2e — functions, geometry and probability

This lesson is maintained against the cited sources below. Simulations are learning models and do not replace supervised physical-laboratory safety procedures.

Content publisher
WebLab STEM
Technical maintainer
Đức Tiến — Control & Automation Engineer