What is a probability distribution in the context of IB DP Mathematics?

In IB DP Mathematics, a probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment. It is a fundamental concept in probability theory, used to describe the likelihood of various outcomes. Probability distributions can be discrete or continuous, depending on whether they deal with discrete or continuous random variables.

How do discrete and continuous probability distributions differ?

Discrete probability distributions apply to scenarios where the set of possible outcomes is countable, such as rolling a die or flipping a coin. Examples include the binomial and Poisson distributions. Continuous probability distributions, on the other hand, apply to scenarios where outcomes are not countable and can take any value within a range, such as measuring height or temperature. The normal distribution is a common example of a continuous distribution.

What is the significance of the normal distribution in probability?

The normal distribution, also known as the Gaussian distribution, is significant in probability because it describes how values of a variable are distributed. It is characterized by its bell-shaped curve and is defined by its mean and standard deviation. The normal distribution is important in the IB DP curriculum because it models many natural phenomena and is used in various statistical methods, including hypothesis testing and confidence intervals.

How is the probability mass function (PMF) used in probability distributions?

The probability mass function (PMF) is used in discrete probability distributions to specify the probability of each possible outcome. It assigns a probability to each value of a discrete random variable. The PMF is a crucial concept in IB DP Mathematics as it helps in calculating probabilities and understanding the distribution of discrete random variables, such as in the binomial distribution.

What role does the cumulative distribution function (CDF) play in probability theory?

The cumulative distribution function (CDF) plays a vital role in probability theory by providing the probability that a random variable takes on a value less than or equal to a specific value. It is applicable to both discrete and continuous probability distributions. In the IB DP curriculum, understanding the CDF is essential for analyzing the behavior of random variables and for solving problems related to probability and statistics.

Why is "Computing probabilities without writing $X \sim B(n, p)$ or $X \sim N(\mu, \sigma^2)$." a common mistake in Probability Distribution?

Why it happens: Jumping to the GDC. How to avoid it: STATE the distribution and parameters — this earns AO1 method marks even if the arithmetic slips.

Why is "Using variance where the GDC requires standard deviation." a common mistake in Probability Distribution?

Why it happens: N(, ^2) has ^2 but GDC takes . How to avoid it: N(170, 64) = 8, NOT 64.

Why is "$P(X \ge 7) = 1 - $ binomCdf$(n, p, 7)$ for a discrete variable." a common mistake in Probability Distribution?

Why it happens: Off-by-one — Cdf(n, p, 7) = P(X 7). How to avoid it: P(X 7) = 1 - P(X 6) = 1 - binomCdf(n, p, 6).

Probability Distribution – Study Notes & Past Paper Style Questions | IB Maths AA HL

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Probability Distributions — IB Maths AA HL: discrete and continuous, $E(X)$, $\text{Var}(X)$, binomial, normal

AA HL extends SL with continuous random variables (probability density functions) and richer normal-distribution applications. This note covers discrete and continuous distributions, expected value, variance, and the two named distributions $B(n, p)$ and $N(\mu, \sigma^2)$ .

At a glance

DISCRETE: probabilities sum to 1; $E(X) = \sum x P(X = x)$ ; $\text{Var}(X) = E(X^2) - [E(X)]^2$ .

What you’ll learn

Mapped to the IB DP Maths AA HL subject guide (2021 onwards (applies to 2026 exams)).

AO1 — Define discrete and continuous random variables; state expected value formulas.
AO1 — Apply binomial and normal formulas.
AO2 — Compute probabilities and inverse normals via GDC.
AO2 — Find $E(X)$ and $\text{Var}(X)$ for continuous distributions via integration.
AO3 — Justify the choice of distribution with the four binomial conditions or normal context.

Discrete and continuous random variables

Sum or integrate.

Discrete. $X$ takes values $x_1, x_2, \ldots$ with probabilities summing to 1.

Binomial and Normal distributions

$B(n, p)$ for counts, $N(\mu, \sigma^2)$ for measurements.

Binomial $X \sim B (n,$ . independent trials, each success with probability .

Quick recap

Discrete: $\sum p = 1$ , $E(X) = \sum xp$ , $Var = E (X^{2}) - [E (X)$ .

Memorise this

Verbatim phrases, formulae and definitions IB DP mark schemes credit (key for AO1 knowledge marks on Paper 1).

$E(X) = \sum xp$ (discrete) or $\int xf$ (continuous)
$Var (X) = E (X^{2}) -$

How it’s examined

Paper 1: discrete distribution tables, small- $n$ binomial, exact normal values via standardisation. Paper 2: GDC arithmetic. Paper 3: extended-response inference and modelling.

Sources: IB Diploma Programme Mathematics: Analysis and Approaches subject guide (IBO, official). Last reviewed 2026-05-31.

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\text{Var} = E(X^2) - [E(X)]^2

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\text{Var}(X) = E(X^2) - [E(X)]^2

Probability Distribution – Study Notes & Past Paper Style Questions | IB Maths AA HL | Tutopiya

Probability Distribution

Detailed Study Notes

Take this whole topic with you

Probability Distribution — frequently asked questions

Probability Distributions — IB Maths AA HL: discrete and continuous, $E(X)$, $\text{Var}(X)$, binomial, normal

At a glance

What you’ll learn

Discrete and continuous random variables

Binomial and Normal distributions

Quick recap

Memorise this

How it’s examined

Step-by-step worked examples — Probability Distribution

1Expected value from a table

2Binomial — at least one

3Normal — central probability and cutoff

4Continuous: verify pdf and find variance

Key Definitions and Keywords — Probability Distribution

Binomial distribution $B(n, p)$

Normal distribution $N(\mu, \sigma^2)$

Probability density function (pdf)

Common Mistakes and Misconceptions — Probability Distribution

✕Computing probabilities without writing $X \sim B(n, p)$ or $X \sim N(\mu, \sigma^2)$ .

✕Using variance where the GDC requires standard deviation.

✕ $P(X \ge 7) = 1 -$ binomCdf $(n, p, 7)$ for a discrete variable.

Probability Distribution

Detailed Study Notes

Take this whole topic with you

Probability Distribution — frequently asked questions

At a glance

What you’ll learn

Quick recap

Memorise this

How it’s examined

1Expected value from a table

2Binomial — at least one

3Normal — central probability and cutoff

4Continuous: verify pdf and find variance

Binomial distribution B(n,p)B(n, p)B(n,p)

Normal distribution N(μ,σ2)N(\mu, \sigma^2)N(μ,σ2)

Probability density function (pdf)

✕Computing probabilities without writing X∼B(n,p)X \sim B(n, p)X∼B(n,p) or X∼N(μ,σ2)X \sim N(\mu, \sigma^2)X∼N(μ,σ2).

✕Using variance where the GDC requires standard deviation.

✕P(X≥7)=1−P(X \ge 7) = 1 - P(X≥7)=1− binomCdf(n,p,7)(n, p, 7)(n,p,7) for a discrete variable.

Binomial distribution $B(n, p)$

Normal distribution $N(\mu, \sigma^2)$

✕Computing probabilities without writing $X \sim B(n, p)$ or $X \sim N(\mu, \sigma^2)$ .

✕ $P(X \ge 7) = 1 -$ binomCdf $(n, p, 7)$ for a discrete variable.