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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) = P(F∩G)P(G)\frac{P(F \cap G)}{P(G)}

Step 2: P(F ∩ G) = 0.20 (given)

Step 3: P(F|G) = 0.200.40=0.50\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∼N(50,16)X \sim N(50, 16). Find P(X > 55).

Solution: Step 1: Standardise: Z=X−μσ=55−504=1.25Z = \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∼B(8,0.4)X \sim B(8, 0.4). Find P(X = 3).

Solution: Step 1: Formula: P(X=k)=(nk)pk(1−p)n−kP(X = k) = \binom{n}{k} p^k (1-p)^{n-k}

Step 2: P(X=3)=(83)(0.4)3(0.6)5P(X = 3) = \binom{8}{3} (0.4)^3 (0.6)^5

Step 3: (83)=8!3!5!=56\binom{8}{3} = \frac{8!}{3!5!} = 56

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

Key insight: The binomial coefficient (nk)\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