$X \sim B(40, 0.75)$. Find $P(28 \le X < 35)$ to 3 s.f.

1. Translate : $P(28 \le X < 35)$. 2. Discrete upper bound : $X < 35 \Rightarrow X \le 34$. 3. Target : $P(28 \le X \le 34)$. 4. Tool : BinomialCD, Lower $=28$, Upper $=34$, $n=40$, $p=0.75$. 5. Execute : $0.803734\ldots \Rightarrow \mathbf{0.804}$.

IB Maths AI SL · Unit 4: Statistics and Probability

Normal & Binomial Distributions (IB Maths AI SL): notes, exam questions and worked examples

Updated October 2026 · By Pete Bromfield

Use the binomial distribution $X \sim B (n, p)$ for a fixed number of independent trials with the same chance of success, and the normal distribution $X \sim N (μ, σ^{2})$ for a continuous, symmetric, bell-shaped variable such as heights or times. In AI SL both are done on your GDC: write the distribution, the probability statement and the answer.

29 questions
Paper 1: 8
Paper 2: 21
3 worked examples

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Notes
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Worked examples
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Normal & Binomial Distributions notes

The binomial distribution

A binomial model needs a fixed number of trials $n$ , two outcomes, a constant probability $p$ and independent trials. Then $E (X) = n p$ . If 20% of seeds fail and you plant 10, $X \sim B (10, 0.2)$ and you expect $2$ to fail.

Binomial probabilities on the GDC

Binomial pdf gives $P (X = r)$ ; binomial cdf gives $P (X \leq r)$ . Turn other inequalities into these: $P (X \geq 3) = 1 - P (X \leq 2)$ and $P (X < 3) = P (X \leq 2)$ . With $X \sim B (10, 0.2)$ , $P (X = 2) \approx 0.302$ .

The normal distribution

The curve is symmetric about the mean $μ$ , and the spread is set by the standard deviation $σ$ . About 68% of values lie within one standard deviation of the mean, 95% within two and 99.7% within three. Sketch the curve and shade the region before you calculate.

Normal probabilities and inverse normal

Normal cdf takes a lower bound, an upper bound, $μ$ and $σ$ . If test scores are $N (60, 12^{2})$ , $P (X > 75) \approx 0.106$ . Inverse normal finds the value for a given area to the left: the top 10% of scores start at about $75.4$ .

Which distribution?

Counting successes in a fixed number of trials: binomial. Measuring a continuous quantity: normal. "At least", "more than" and "at most" decide the inequality, so read them carefully.

Key formulas

Binomial $X \sim B (n, p)$ Learn it: $P (X = r) = (\binom{n}{r}) p^{r} (1 - p)^{n - r}$
Binomial mean, variance In the booklet: $E (X) = n p, Var (X) = n p (1 - p)$
Normal: about 68%, 95%, 99.7% of values lie within 1, 2, 3 s.d. of $μ$ Learn it
Expected value In the booklet: $E (X) = \sum x P (X = x)$

The IB formula booklet is provided in every exam paper. In the booklet = printed there, so know where to find it; Learn it = not in the booklet, so learn it by heart. All AI SL formulas →

Explained, with a worked example: Binomial distribution

Normal & Binomial Distributions worked examples

Worked example 1: Normal distribution probability · easy

Suitcase weights $W \sim N (20, {3.5}^{2})$ kg. Find $P (W < 15)$ to 3 s.f.

Solution

5 steps · 1 to fill in

Examiner tip: For continuous distributions,

P (W < 15) = P (W \leq 15)

P (W = 15)

is exactly zero.

Worked example 2: Inverse normal — top 10% mark · medium

Exam marks $\sim N (300, 100^{2})$ . The top $10 %$ receive a prize. Find the minimum qualifying mark (nearest integer).

Solution

5 steps · 1 to fill in

Examiner tip: If your calculator only accepts left-tail areas, convert top percentages via

1 - p

(top

10 % \Rightarrow 0.90

Worked example 3: Binomial distribution — range of successes · hard

$X \sim B (40, 0.75)$ . Find $P (28 \leq X < 35)$ to 3 s.f.

Solution

5 steps · 1 to fill in

Examiner tip: Unlike the Normal distribution, strict inequalities matter for the Binomial — "fewer than 35" strictly means at most 34.

Normal & Binomial Distributions exam-style questions

3 questions from our AI SL bank, written in the style of IB papers, easiest first. One has its full worked solution open here. Open the worked answer, or check your answer in the practice area.

Question 1 · easy · 3 marks · Paper 1

The probability that a flight arrives on time is 0.9. An airline records the arrivals of 75 flights. Calculate the probability that strictly more than 72 flights arrive on time.

Full worked solution Free, no sign-in

Let $X$ be the number of flights on time: $X \sim B (75, 0.9)$ M1
"Strictly more than 72" means $P (X > 72) = P (X \geq 73) = 1 - P (X \leq 72)$ (M1)
Using the binomial CDF on the GDC: $P (X \geq 73) = 0.016128 \dots \approx 0.0161$ A1

Every question here has a full worked solution. See them all →

Question 2 · very hard · 7 marks · Paper 2

The weights, $W$ kg, of parcels delivered by a courier are normally distributed with mean $3.8$ kg and standard deviation $1.2$ kg.

Find the probability that a randomly chosen parcel weighs less than $1.5$ kg. [2 marks]
The heaviest $8 %$ of parcels are charged a surcharge. Find the minimum weight of a parcel that is charged the surcharge. [2 marks]
On one day the courier delivers $50$ parcels. Find the expected number of these parcels that weigh less than $1.5$ kg. [1 mark]
A random sample of $20$ parcels is chosen. Find the probability that exactly $3$ of them weigh less than $1.5$ kg. [2 marks]

Check your answer in practice →

Question 3 · very hard · 6 marks · Paper 2

The masses of harvested apples are normally distributed with mean $150$ g and standard deviation $20$ g.

Find the probability that a randomly selected apple has a mass between $140$ g and $160$ g. [2 marks]
The heaviest $10 %$ of apples are sold as "premium". Find the minimum mass of a premium apple. [2 marks]
Two apples are chosen at random. Find the probability that both of them are premium. [2 marks]

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Free account — sign in to download Topic PDF: 7 questions with worked answers · 5 pages · 196 KB

All 29 normal & binomial distributions questions →

Where students lose marks on normal & binomial distributions

Examiner's noteBinomial and normal: recognising the model, the bound, the tail

Where marks are lost: Candidates did not recognise a binomial situation (or its assumptions), used the wrong bound for "at least" or "fewer than", did not know E(X) = np although it is in the formula booklet, assumed an expected number must be a whole number, shaded or used the wrong tail of a normal distribution, and used the complementary area in the inverse normal.

Examiners flagged this in 3 exam series (May 2024, November 2024, November 2025). Summarised in our own words from IB subject reports, IB Programme Resource Centre. Read the full note →

Normal & Binomial Distributions on your calculator

A GDC is allowed on every AI paper. Keystrokes for the Casio fx-CG50, TI-84 Plus CE and TI-Nspire, with a checked worked example:

Normal & Binomial Distributions FAQ

When do I use the binomial distribution instead of the normal?

Use the binomial for a count of successes in a fixed number of independent trials with the same probability each time. Use the normal for a continuous measurement, such as a mass or a time, that is symmetric about its mean.

How do I find P(X ≥ 3) for a binomial on the GDC?

Binomial cdf gives P(X ≤ r), so use P(X ≥ 3) = 1 − P(X ≤ 2).

How many IB Maths AI SL probability distributions questions are there?

There are 29 exam-style probability distributions questions in the AI SL question bank (Paper 1: 8 · Paper 2: 21), graded 5 very hard, 7 hard, 13 medium, 4 easy. Every question has a full IB-style mark scheme (M, A and R marks).

Is probability distributions on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 8 · Paper 2: 21. Practise with your GDC — AI papers expect calculator methods throughout.

Next steps

More AI SL Unit 4: Correlation & Regression · Descriptive Statistics · Probability · Sampling Techniques · Two-Sample t-Test · Venn Diagrams & Hard Probability

Teaching this topic? Free ready-to-teach lessons (slides and teacher notes): Binomial distribution · Continuous distributions (normal on GDC).

The ideas behind it, in the notes: Conditional probability with tree diagrams

Keep revising AI SL: AI SL revision notes · IB Maths AI SL past papers · AI SL common exam mistakes · AI SL syllabus checklist · AI SL statistics and probability knowledge organiser

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Normal & Binomial Distributions (IB Maths AI SL): notes, exam questions and worked examples

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Normal & Binomial Distributions notes

The binomial distribution

Binomial probabilities on the GDC

The normal distribution

Normal probabilities and inverse normal

Which distribution?

Key formulas

Normal & Binomial Distributions worked examples

Worked example 1: Normal distribution probability · easy

Solution

Worked example 2: Inverse normal — top 10% mark · medium

Solution

Worked example 3: Binomial distribution — range of successes · hard

Solution

Normal & Binomial Distributions exam-style questions

Question 1 · easy · 3 marks · Paper 1

Full worked solution Free, no sign-in

Question 2 · very hard · 7 marks · Paper 2

Question 3 · very hard · 6 marks · Paper 2

Where students lose marks on normal & binomial distributions

Normal & Binomial Distributions on your calculator

Normal & Binomial Distributions FAQ

Next steps

What else is free here