Unit 4 · Topic 4: Statistics and Probability
The Complete IB Mathematics: Applications and Interpretation Guide · IB Mathematics: Applications and InterpretationUnit 4 · Topic 4: Statistics and Probability
- The Complete IB Mathematics: Applications and Interpretation Guide
- IB Mathematics: Applications and Interpretation
- 5 sections
International Baccalaureate does not publish a per-unit weighting for this exam. Exam structure (official): SL: Paper 1 (90 min, 40%) · Paper 2 (90 min, 40%) · Internal Assessment (20%); HL: Paper 1 (120 min, 30%) · Paper 2 (120 min, 30%) · Paper 3 (60 min, 2 questions, 20%) · Internal Assessment (20%). Be ready to carry the same unit skills and representations into a complete answer on every paper. This topic develops data, probability, distributions, inference, and regression. The official subject brief names the topic but does not publish a topic-by-topic examination percentage, so no percentage is claimed here on the “Statistics and Probability: choose, calculate, interpret” page.
- How IB Mathematics: Applications and Interpretation assesses this International Baccalaureate publishes no per-unit percentage for IB Mathematics: Applications and Interpretation; Paper 1 and Paper 2 each account for 40% of the SL total and 30% of the HL total. · SL: Paper 1 (90 min, 40%) · Paper 2 (90 min, 40%) · Internal Assessment (20%) | HL: Paper 1 (120 min, 30%) · Paper 2 (120 min, 30%) · Paper 3 (60 min, 2 questions, 20%) · Internal Assessment (20%) · show the model with a probability distribution with a defined random variable, a test statement with hypotheses, test statistic, and p-value, a confidence interval with conditions and a contextual conclusion
- Key skills Sampling, Data Presentation, and Measures of Center and Spread, Pearson's and Spearman's Correlation and Regression, Probability: Combined Events, Conditional Probability, and Expected Outcomes, Discrete Random Variables, Binomial and Normal Distributions, Hypothesis Testing: Chi-Squared Tests and the t-Test, Combinations of Random Variables, the Poisson Distribution, the Central Limit Theorem, and Confidence Intervals (HL), Data Collection Design, Further Hypothesis Tests, and Type I and II Errors (HL), Markov Chains and Transition Matrices (HL)
- How to study for Unit 4 This page turns a probability distribution with a defined random variable, a test statement with hypotheses, test statistic, and p-value, a confidence interval with conditions and a contextual conclusion into one route: State the random variable or population parameter and check the required conditions before entering values into technology..
- The organizing decision Define the population, parameter, or random variable, verify the model conditions, and choose the probability or inference method.
What IB Mathematics: Applications and Interpretation Unit 4 covers
Use this map to connect each assessed skill to the relationship or representation that makes it visible.
Sampling, Data Presentation, and Measures of Center and Spread
random variable; p-value; confidence interval02Pearson's and Spearman's Correlation and Regression
random variable; p-value; confidence interval03Probability: Combined Events, Conditional Probability, and Expected Outcomes
random variable; p-value; confidence interval04Discrete Random Variables, Binomial and Normal Distributions
random variable; p-value; confidence interval05Hypothesis Testing: Chi-Squared Tests and the t-Test
random variable; p-value; confidence interval06Combinations of Random Variables, the Poisson Distribution, the Central Limit Theorem, and Confidence Intervals (HL)
random variable; p-value; confidence interval07Data Collection Design, Further Hypothesis Tests, and Type I and II Errors (HL)
random variable; p-value; confidence interval08Markov Chains and Transition Matrices (HL)
random variable; p-value; confidence intervalInternational Baccalaureate does not publish a per-unit weighting for this exam.
Official unit name and weighting: IB subject guide / subject brief.
Statistics and Probability: choose, calculate, interpret
Connect the published share to the unit model
This topic develops data, probability, distributions, inference, and regression. The official subject brief names the topic but does not publish a topic-by-topic examination percentage, so no percentage is claimed here on the “Statistics and Probability: choose, calculate, interpret” page.
Vocabulary in use: population, sample, random variable, expected value, standard deviation, null hypothesis, p-value, confidence interval. Each term is applied on these pages and collected again in the glossary with a checked page pointer on the “Statistics and Probability: choose, calculate, interpret” page.
A strong statistics solution defines the random variable or population parameter, verifies conditions, preserves precision, and states the inference in context. The sensor, game, refill, and battery data in this topic are independently written illustrations.
The decision that organizes this unit
Define the system and choose the route before calculating
Define the population, parameter, or random variable, verify the model conditions, and choose the probability or inference method.
State the random variable or population parameter and check the required conditions before entering values into technology.
Mechanism route and repair branches
Relationships to preserve
- A binomial model combines a fixed trial count with two outcomes, constant probability, and independence.
- Expected value weights every exhaustive outcome by its probability and describes a long-run average.
- A hypothesis test compares a p-value with the significance level, while a confidence interval estimates a population parameter.
Representations to read
- a probability distribution with a defined random variable
- a test statement with hypotheses, test statistic, and p-value
- a confidence interval with conditions and a contextual conclusion
Branches to reject
- using a binomial calculation without checking fixed trials, two outcomes, constant probability, and independence
- interpreting expected value as the guaranteed result of one play
- giving a p-value or interval without a conclusion about the population in context
| Key concept | Why it's hard | What scores |
|---|---|---|
| population | Distinguishing population from sample requires their definitions and units. | Define the quantity measured by population and attach its units before calculation. |
| random variable | A calculator screen does not show why random variable is the suitable model feature. | Record the relation used for random variable and one identifying intermediate value. |
| null hypothesis | A computed null hypothesis value still needs a domain and scale check. | Check the restrictions, precision, and contextual meaning of the null hypothesis result. |
How IB Mathematics: Applications and Interpretation assesses Topic 4: Statistics and Probability
What a complete response must make visible
Match the task to evidence that a reader can audit, then check the most likely reasoning failure before finalizing the response.
| Task | Evidence to show | Hurdle |
|---|---|---|
| Sampling, Data Presentation, and Measures of Center and Spread | random variable; p-value; confidence interval | using a binomial calculation without checking fixed trials, two outcomes, constant probability, and independence |
| Pearson's and Spearman's Correlation and Regression | random variable; p-value; confidence interval | interpreting expected value as the guaranteed result of one play |
| Probability: Combined Events, Conditional Probability, and Expected Outcomes | random variable; p-value; confidence interval | giving a p-value or interval without a conclusion about the population in context |
| Discrete Random Variables, Binomial and Normal Distributions | random variable; p-value; confidence interval | using a binomial calculation without checking fixed trials, two outcomes, constant probability, and independence |
| Hypothesis Testing: Chi-Squared Tests and the t-Test | random variable; p-value; confidence interval | interpreting expected value as the guaranteed result of one play |
| Combinations of Random Variables, the Poisson Distribution, the Central Limit Theorem, and Confidence Intervals (HL) | random variable; p-value; confidence interval | giving a p-value or interval without a conclusion about the population in context |
| Data Collection Design, Further Hypothesis Tests, and Type I and II Errors (HL) | random variable; p-value; confidence interval | using a binomial calculation without checking fixed trials, two outcomes, constant probability, and independence |
| Markov Chains and Transition Matrices (HL) | random variable; p-value; confidence interval | interpreting expected value as the guaranteed result of one play |
Build a binomial probability model
Read each question as it appears on the exam paper, choose your answer, then tap it to see the feedback.
Practice 1: Build an exact binomial probability
A new sensor independently flags a package with probability 0.18 on each of 12 controlled trials.
Find the probability that the sensor flags exactly three packages.
State why a binomial model is suitable.
Show your calculation.
A. P(X = 3) = C(12,3)(0.18)³(0.82)⁹ ≈ 0.215 after checking fixed trials, two outcomes, constant probability and independence.
✅ Correct! 😊 Correct. The four binomial conditions are addressed and the term uses three flags with nine non-flags.
B. P(X = 3) = C(12,3)(0.18)³(0.18)⁹ because every trial has flag probability 0.18.
❌ Not quite 😭 The nine non-flag outcomes require probability 0.82. Using 0.18 for both outcomes gives the wrong binomial term.
Prep with Sia Agent
Free download for Mac and Windows. Choose your computer:
C. P(X = 3) = (0.18)³ ≈ 0.00583 because only the three flagged packages matter.
❌ Not quite 😭 This omits the arrangements of the three flags and the nine non-flag probabilities, so it does not represent exactly three among twelve.
Prep with Sia Agent
Free download for Mac and Windows. Choose your computer:
D. Use P(X ≤ 3) because ‘exactly three’ includes zero, one and two flags.
❌ Not quite 😭 A cumulative probability answers at most three. The wording asks only for X = 3.
Prep with Sia Agent
Free download for Mac and Windows. Choose your computer:
Full explanation
- Step 1Let X be the number of flags in 12 trials. The trials are fixed, independent, have two outcomes and constant p = 0.18.
- Step 2For exactly three flags, choose three trial positions and use nine complementary outcomes.
- Step 3P(X = 3) = C(12,3)(0.18)³(0.82)⁹.
- Step 4Evaluation gives approximately 0.215 for exactly three flags in this controlled process.
Practice 2: Interpret expected value
A game gives a net gain of 10 dollars with probability 0.30 and a net loss of 5 dollars with probability 0.70.
Find the expected net result per play and interpret your answer.
Show your working.
A. E(X) = (10 − 5) ÷ 2 = 2.50 dollars, so the game gains 2.50 dollars per play.
❌ Not quite 😭 This averages the two outcomes without their unequal probabilities, so it is not the expected value.
Prep with Sia Agent
Free download for Mac and Windows. Choose your computer:
B. E(X) = 10(0.30) + 5(0.70) = 6.50 dollars because both amounts are positive sizes.
❌ Not quite 😭 A loss is a negative net result. Treating 5 dollars as positive reverses its contribution to expectation.
Prep with Sia Agent
Free download for Mac and Windows. Choose your computer:
C. E(X) = 10(0.30) + (−5)(0.70) = −0.50 dollars; over many plays the mean loss is 0.50 dollars per play.
✅ Correct! 😊 Correct. The outcomes are weighted by exhaustive probabilities, and the negative result is interpreted as a long-run average loss.
D. E(X) = −0.50 dollars, so every individual play loses exactly 0.50 dollars.
❌ Not quite 😭 The calculation is right, but expected value is a long-run mean and is not a guaranteed outcome on one play.
Prep with Sia Agent
Free download for Mac and Windows. Choose your computer:
Full explanation
- Step 1Let X be the net result; its possible values are 10 and −5 dollars, with probabilities summing to 1.
- Step 2Weight each outcome: E(X) = 10(0.30) + (−5)(0.70).
- Step 3Calculate 3.00 − 3.50 = −0.50 dollars.
- Step 4The negative expectation represents a long-run mean loss of 0.50 dollars per play, not each single outcome.
Key terms for Unit 4: Topic 4: Statistics and Probability
Models, uses, and boundaries
- Form a Binomial Probability
- A binomial term combines a trial count, success probability, and complementary probability for one named outcome count. Use this relation when the named quantities and conditions in the question match the calculation described here.
- Calculate an Exact Binomial Probability
- Evaluate C(12,3)(0.18)^3(0.82)^9 for exactly three flags; check fixed trials, two outcomes, constant probability, and independence before calculation. Use this relation when the named quantities and conditions in the question match the calculation described here.
- Find Expected Value
- Calculate E(X) = 8(0.25) + (−3)(0.75) = −0.25 dollars; verify that the two outcome probabilities are exhaustive and sum to 1. Use this relation when the named quantities and conditions in the question match the calculation described here.
- Complete a Two-Sided Mean Test
- Compute t = (52.4 − 55)/(6/√16) = −1.733 with 15 degrees of freedom; check independent selection, the under-10% condition, and the absence of strong skew or outliers. Use this relation when the named quantities and conditions in the question match the calculation described here.
- Construct a Mean Confidence Interval
- Calculate SE = 0.5 h and margin 1.032 h to form 17.568 to 19.632 h; check independent sampling, production above 250, and a roughly symmetric plot without outliers. Use this relation when the named quantities and conditions in the question match the calculation described here.
IB Mathematics: Applications and Interpretation Unit 4 FAQ
How much of IB Mathematics: Applications and Interpretation does Unit 4 carry?
International Baccalaureate does not publish a per-unit weighting for this exam.
What is the first move on a Topic 4: Statistics and Probability problem?
State the random variable or population parameter and check the required conditions before entering values into technology.
Which relationships should I preserve?
A binomial model combines a fixed trial count with two outcomes, constant probability, and independence. Expected value weights every exhaustive outcome by its probability and describes a long-run average. A hypothesis test compares a p-value with the significance level, while a confidence interval estimates a population parameter.
Which representations should I practice?
Practice moving among a probability distribution with a defined random variable, a test statement with hypotheses, test statistic, and p-value, a confidence interval with conditions and a contextual conclusion.
What error should I check before submitting an answer?
Check for using a binomial calculation without checking fixed trials, two outcomes, constant probability, and independence; interpreting expected value as the guaranteed result of one play; giving a p-value or interval without a conclusion about the population in context.
Evidence workshop
Continue from the free model into complete practice
The full unit guide continues with the chapter’s worked examples, figures, scoring tables, and answer checks.
- Build a binomial probability model
- Calculate and interpret expected value
- Complete a two-sided mean test
- Check and answer: Build a binomial probability model · Complete a two-sided mean test
- Construct a mean confidence interval
- Check and answer: Calculate and interpret expected value · Construct a mean confidence interval
- Check the binomial model and mean test
- Check expected value and the confidence interval
Full unit practice. Open the complete guide for the full evidence workshop and synthesis.
Related IB Mathematics: Applications and Interpretation unit guides
IB Mathematics: Applications and Interpretation Exam Guide & Review
The whole exam and its official unit sequence.01Topic 1: Number and Algebra
International Baccalaureate does not publish a per-unit weighting for this exam.02Topic 2: Functions
International Baccalaureate does not publish a per-unit weighting for this exam.03Topic 3: Geometry and Trigonometry
International Baccalaureate does not publish a per-unit weighting for this exam.05Topic 5: Calculus
International Baccalaureate does not publish a per-unit weighting for this exam.How to study IB Mathematics: Applications and Interpretation Unit 4
Start with the organizing decision
Before solving, restate the decision in operational terms: Define the population, parameter, or random variable, verify the model conditions, and choose the probability or inference method.. Your first written move should be to State the random variable or population parameter and check the required conditions before entering values into technology..
Practice the same idea in several representations
Rotate through a probability distribution with a defined random variable, a test statement with hypotheses, test statistic, and p-value, a confidence interval with conditions and a contextual conclusion. Use each representation to practice Sampling, Data Presentation, and Measures of Center and Spread, Pearson's and Spearman's Correlation and Regression, Probability: Combined Events, Conditional Probability, and Expected Outcomes, Discrete Random Variables, Binomial and Normal Distributions, Hypothesis Testing: Chi-Squared Tests and the t-Test, Combinations of Random Variables, the Poisson Distribution, the Central Limit Theorem, and Confidence Intervals (HL), Data Collection Design, Further Hypothesis Tests, and Type I and II Errors (HL), Markov Chains and Transition Matrices (HL), and explain what stays invariant when the surface form changes.
Turn each error into a repair check
After every attempt, audit the response for using a binomial calculation without checking fixed trials, two outcomes, constant probability, and independence; interpreting expected value as the guaranteed result of one play; giving a p-value or interval without a conclusion about the population in context. Then redo only the first step that made the reasoning diverge, keeping units, direction, and model conditions visible.
Confirm current course details in the official IB subject guide / subject brief for the 2027 administration.
Where to go next
Return to the IB Mathematics: Applications and Interpretation to see every unit and plan your revision.