Statistics and Probability Practice (Interactive)

Intuition

Statistics is the science of learning from data — it provides tools to extract meaning from uncertainty and variation: Statistical thinking requires understanding both the mathematics of probability and the practical challenges of data collection and interpretation

Why it matters: In an age of big data, statistical literacy is essential for informed decision-making in every field

The key insight: Statistical thinking requires understanding both the mathematics of probability and the practical challenges of data collection and interpretation

IB Maths — Statistics and Probability Practice

18 auto-graded practice problems covering core IB Maths statistics and probability content, aligned to the SL/HL syllabus. Select an answer, submit, and review the explanation.

Worked Examples

Example 1: Conditional Probability

Problem: In a class, 60% of students study French (F) and 40% study German (G). 20% study both. Find P(F|G).

Solution: Step 1: P(F|G) = $\frac{P(F \cap G)}{P(G)}$

Step 2: P(F ∩ G) = 0.20 (given)

Step 3: P(F|G) = $\frac{0.20}{0.40} = 0.50$

Key insight: Conditional probability is the probability of one event given that another has occurred. The formula P(A|B) = P(A ∩ B) / P(B) always works.

Example 2: Normal Distribution

Problem: $X \sim N(50, 16)$. Find P(X > 55).

Solution: Step 1: Standardise: $Z = \frac{X - \mu}{\sigma} = \frac{55 - 50}{4} = 1.25$

Step 2: P(X > 55) = P(Z > 1.25) = 1 - P(Z ≤ 1.25)

Step 3: From Z-tables: P(Z ≤ 1.25) = 0.8944

Step 4: P(X > 55) = 1 - 0.8944 = 0.1056

Key insight: Always standardise before using Z-tables. The formula Z = (X - μ) / σ converts any normal distribution to the standard normal.

Example 3: Binomial Probability

Problem: $X \sim B(8, 0.4)$. Find P(X = 3).

Solution: Step 1: Formula: $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$

Step 2: $P(X = 3) = \binom{8}{3} (0.4)^3 (0.6)^5$

Step 3: $\binom{8}{3} = \frac{8!}{3!5!} = 56$

Step 4: $P(X = 3) = 56 \times 0.064 \times 0.07776 = 56 \times 0.00498 = 0.2787$

Key insight: The binomial coefficient $\binom{n}{k}$ counts the number of ways to choose k successes from n trials.

Descriptive Statistics

Probability

Binomial Distribution

Normal Distribution

Correlation

Hypothesis Testing

Common Mistakes

Confusing Type I and Type II errors: Type I is rejecting a true null hypothesis (false positive). Type II is failing to reject a false null hypothesis (false negative). Don’t mix up which is which.

Forgetting that correlation does not imply causation: Two variables can be correlated without one causing the other. Always consider confounding variables.

Mixing up population parameters with sample statistics: μ and σ are population parameters. x̄ and s are sample statistics. Use the correct notation and formulas for each.

Cross-References

• Number and Algebra: Algebra supports all practice
• Functions: Functions are tested
• Calculus: Calculus practice covers core techniques

Advanced Content

This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.

Derivations and Proofs

Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.

Extended Examples

Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.

Research Connections

This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.

Prerequisites

Ensure you have mastered the prerequisite material before attempting this advanced content.

Advanced Content

This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.

Derivations and Proofs

Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.

Extended Examples

Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.

Research Connections

This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.

Prerequisites

Ensure you have mastered the prerequisite material before attempting this advanced content.

Advanced Content

This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.

Derivations and Proofs

Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.

Extended Examples

Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.

Research Connections

This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.

Prerequisites

Ensure you have mastered the prerequisite material before attempting this advanced content.

Advanced Content

This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.

Derivations and Proofs

Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.

Extended Examples

Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.

Research Connections

This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.

Prerequisites

Ensure you have mastered the prerequisite material before attempting this advanced content.

Advanced Content

This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.

Derivations and Proofs

Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.

Extended Examples

Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.

Research Connections

This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.

Prerequisites

Ensure you have mastered the prerequisite material before attempting this advanced content.