In IB Diploma Mathematics: Applications and Interpretation (AI), both SL and HL students need to recognize situations that can be modelled by a binomial distribution, use their graphical display calculator (GDC) accurately, and interpret answers in context. This tutorial explains the four conditions, works through a realistic exam-style example, and ends with a short self-check.
When can you use a binomial distribution?
Write X ~ B(n, p) when X counts the number of successes in n trials, provided that: (1) the number of trials is fixed; (2) each trial has exactly two outcomes, success or failure; (3) the probability of success p stays constant; and (4) the trials are independent. A “success” simply means the event you are counting; it need not be something desirable.
For a binomial random variable, P(X = k) = C(n,k) pk(1 − p)n−k, where C(n,k) is the number of ways to choose k successes from n trials. Also, E(X) = np, Var(X) = np(1 − p), and SD(X) = √[np(1 − p)].
Worked example: online mathematics practice
Suppose each of eight independent practice questions has a 30% probability of being answered correctly by a student. Assume this probability remains the same for each question. Let X be the number answered correctly. These assumptions are deliberately simplified for mathematics practice; real student performance may not meet them.
(a) State the distribution of X. (b) Find the probability of exactly two correct answers. (c) Find the probability of at least one correct answer. (d) Find the mean and standard deviation. (e) Explain one reason this model might not work well for an actual student.
Step 1: Identify the model
There are eight trials, each with a correct/incorrect outcome, constant success probability 0.30, and assumed independence. Therefore X ~ B(8, 0.30).
Step 2: Exactly two successes
Use the binomial probability formula: P(X = 2) = C(8,2)(0.30)2(0.70)6 = 28 × 0.09 × 0.117649 = 0.29647548 ≈ 0.296 (3 significant figures). On a GDC, use the binomial probability function with n = 8, p = 0.3, and x = 2. Depending on your calculator, the function may be labelled binompdf or Binomial PD.
Step 3: At least one success
“At least one” means X ≥ 1. Rather than adding eight separate probabilities, use the complement: P(X ≥ 1) = 1 − P(X = 0) = 1 − (0.70)8 = 1 − 0.05764801 = 0.94235199 ≈ 0.942. This is a common exam shortcut. With cumulative technology, you can calculate 1 − P(X ≤ 0).
Step 4: Mean and standard deviation
E(X) = np = 8(0.30) = 2.4 correct answers. Var(X) = np(1 − p) = 8(0.30)(0.70) = 1.68. Hence SD(X) = √1.68 ≈ 1.30 correct answers. The mean 2.4 is an average over many repetitions, not a possible score on a single eight-question quiz.
Step 5: Evaluate the model
Real questions can vary in difficulty, so the probability of a correct answer may not remain at 0.30. Responses may also be dependent if one question teaches the student something useful for the next. Either issue could make the binomial model inappropriate. In an IB exam, state the specific assumption that fails.
GDC and examination techniques
- Define X clearly: “number of correctly answered questions,” not just “number of questions.”
- Translate wording carefully: “exactly 2” means P(X = 2); “at most 2” means P(X ≤ 2); “more than 2” means P(X ≥ 3).
- Choose the right calculator function: use the probability function for one exact value and the cumulative function for an interval up to a value.
- Use complements: P(X ≥ 1) = 1 − P(X = 0) is faster and reduces entry errors.
- Keep precision: retain calculator precision during working, then round the final answer appropriately.
- Interpret your answer: a probability of 0.296 means approximately a 29.6% chance under the stated assumptions—not that 29.6% of the eight questions will be correct.
Self-check: can you apply the method?
A student independently attempts five multiple-choice questions, with probability 0.40 of answering each correctly. Let Y be the number answered correctly. (a) State the distribution of Y. (b) Calculate P(Y = 1). (c) Calculate P(Y ≥ 1). (d) State E(Y).
Answers: (a) Y ~ B(5, 0.40). (b) P(Y = 1) = 5(0.4)(0.6)4 = 0.2592. (c) P(Y ≥ 1) = 1 − 0.65 = 0.92224. (d) E(Y) = 5(0.4) = 2. If you got a different answer for (c), check whether you accidentally calculated “exactly one” instead of “at least one.”
Further IB Mathematics support
For course-specific help, explore our IB Mathematics AI SL and HL tutoring and online IB Mathematics tutoring for students in Europe. You can also book a complimentary first 80-minute mathematics lesson to discuss your mathematics learning goals. This free first lesson is for mathematics tutoring only.
Curriculum reference: International Baccalaureate Diploma Programme mathematics. This independently created example is not an official IB examination question.