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IB Maths AI SL · Unit 4: Statistics and Probability

IB Maths AI SL Probability Distributions Questions

Exam-style IB Maths AI SL probability distributions questions with worked solutions. Start with the 3 fully worked examples below — each is solved step by step the way an IB examiner expects — then try the practice questions and check your working against the mark scheme.

Practise Probability Distributions questions → AI SL formula booklet

What you need to know

P(A|B) = P(A ∩ B)/P(B) — but SL AI expects you to build the tree diagram and read off the answer, not manipulate the formula abstractly. Conditional probability with tree diagrams overview →

Perhaps SL AI's single most examined topic. Know the difference between GoF and independence tests, calculate expected values, and interpret the p-value against the significance level. Chi-squared test — goodness of fit and independence overview →

What's examined in AI SL probability distributions

The question bank covers these probability distributions question types (number of questions in brackets):

Key formulas

Probability of an event
\(P(A) = \dfrac{n(A)}{n(U)}\)
Conditional probability
\(P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}\)
Independent events
\(P(A \cap B) = P(A)\,P(B)\)
Complementary events
\(P(A') = 1 - P(A)\)
Combined events
\(P(A \cup B) = P(A) + P(B) - P(A \cap B)\)
Mutually exclusive events
\(P(A \cap B) = 0\)

In the same notation as the IB formula booklet. All AI SL formulas →

Probability 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

1. Statement: $P(W < 15)$.

2. Tool: NormCD on the GDC.

3. Inputs: Lower $=-10^{99}$, Upper $=15$, $\sigma=3.5$, $\mu=20$.

4. Execute: $0.076563\ldots$

5. State: $\mathbf{0.0766}$.

Examiner tip: For continuous distributions, $P(W<15) = P(W\le 15)$. $P(W=15)$ 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

1. Recognise: top $10\%$ = right tail area $0.10$.

2. Left area: $1 - 0.10 = 0.90$.

3. Tool: InvNorm on the GDC.

4. Inputs: Area $=0.90$ (left), $\sigma=100$, $\mu=300$.

5. Execute: $k = 428.155\ldots \Rightarrow \mathbf{428}$.

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 $P(28 \le X < 35)$ to 3 s.f.

Solution

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: $0.803734\ldots \Rightarrow \mathbf{0.804}$.

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

Try these IB Maths AI SL probability distributions questions

Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area.

Question 1 · medium · 4 marks · Paper 2

A biased die has a probability of \(0.4\) of landing on a six. The die is rolled 15 times. Let \(X\) be the number of sixes rolled.

  1. Identify the distribution of \(X\), including its parameters.
  2. Calculate the probability that the die lands on a six at least 6 times.
Attempt it and see the mark scheme →

Question 2 · hard · 3 marks · Paper 2

A biased coin has a probability of landing on Heads of \(0.6\). The coin is flipped 12 times. Let \(X\) be the number of times the coin lands on Heads.

Find the exact probability that the coin lands on Heads exactly 7 times.

Attempt it and see the mark scheme →

Question 3 · 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.

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

All 18 probability distributions questions with mark schemes →

FAQ

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

There are 18 exam-style probability distributions questions in the AI SL question bank (Paper 1: 6 · Paper 2: 12), graded 3 very hard, 3 hard, 6 medium, 6 starter. 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: 6 · Paper 2: 12. Practise with your GDC — AI papers expect calculator methods throughout.

Where can I get the mark schemes?

Open the AI SL Unit 4 practice page: every question has a step-by-step IB-style mark scheme, and you can photograph your working for instant AI marking. The worked examples on this page are free.

More AI SL Unit 4 topics

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