Discrete probability distributions

IB Mathematics: Applications and Interpretation SL· 30 min read

1. Discrete Random Variables and Valid Probability Distributions★★☆☆☆⏱ 15 min

📘 Definition

Discrete Probability Distribution

P(X=x)

A function that assigns a probability to every possible outcome of a discrete random variable, satisfying two core conditions: all probabilities are between 0 and 1, and the sum of all probabilities equals 1.

Example:

Number of rainy days in a week, number of defective products in a batch

IB exams very commonly ask you to find an unknown probability in a given distribution. You will always use the core rule that the sum of all probabilities equals 1 to solve for the unknown value, then confirm it meets the 0 ≤ p ≤ 1 condition.

📐 Worked Example

A discrete random variable $X$ has the probability distribution: $P (X = 0) = 0.12$ , $P (X = 1) = k$ , $P (X = 2) = 0.45$ , $P (X = 3) = 0.28$ . Find the value of $k$ and confirm the distribution is valid.

1
Use the rule that the sum of all probabilities equals 1:
2
$0.12 + k + 0.45 + 0.28 = 1$
3
Simplify and solve for $k$ :
4
$0.85 + k = 1 \implies k = 1 - 0.85 = 0.15$
5
Check validity: $0 \leq 0.15 \leq 1$ , so $k = 0.15$ gives a valid discrete probability distribution.

2. Expected Value (Mean) of a Discrete Distribution★★☆☆☆⏱ 15 min

📘 Definition

Expected Value

E[X] \text{ or } \mu

The long-term average outcome of many repeated trials of the random variable, calculated as the weighted average of all possible outcomes weighted by their probabilities.

Example:

Expected daily sales, expected profit from a game

The formula for expected value is: $E [X] = \sum x \cdot P (X = x)$ for all possible values of $x$ . Note that the expected value does not need to be one of the possible outcomes of the distribution — it is just an average.

📐 Worked Example

A small shop sells ice creams with the following probability distribution for daily sales. Calculate the expected number of ice creams sold per day:

$x$ (Number of ice creams)	10	15	20	25	30
$P (X = x)$	0.1	0.2	0.35	0.25	0.1

1
Apply the expected value formula:
2
$E[X] = (10 \times 0.1) + (15 \times 0.2) + (20 \times 0.35) + (25 \times 0.25) + (30 \times 0.1)$
3
Calculate each product and sum the results:
4
$E[X] = 1 + 3 + 7 + 6.25 + 3 = 20.25$

3. Variance and Standard Deviation★★★☆☆⏱ 20 min

📘 Definition

Variance

\text{Var}(X) \text{ or } \sigma^2

A measure of the spread of outcomes around the expected value. A higher variance means more variability in outcomes, while a lower variance means outcomes cluster close to the mean.

Example:

A variance of 0 means the outcome is fixed, no variation

The most commonly used formula for variance in IB AI SL is $Var (X) = E [X^{2}] - (E [X])^{2}$ , where $E [X^{2}] = \sum x^{2} \cdot P (X = x)$ . Standard deviation is simply the square root of variance, and is in the same units as the original random variable.

📐 Worked Example

Given $E [X] = 20.25$ for the ice cream sales example above, calculate the variance and standard deviation of daily sales.

1
First calculate $E [X^{2}] = \sum x^{2} P (X = x)$ :
2
$E[X^2] = (10^2 \times 0.1) + (15^2 \times 0.2) + (20^2 \times 0.35) + (25^2 \times 0.25) + (30^2 \times 0.1)$
3
Simplify to get:
4
$E[X^2] = 10 + 45 + 140 + 156.25 + 90 = 441.25$
5
Apply the variance formula:
6
$\text{Var}(X) = 441.25 - (20.25)^2 = 441.25 - 410.0625 = 31.1875$
7
Calculate standard deviation:
8
$\sigma = \sqrt{31.1875} \approx 5.58$

4. Common Pitfalls

Wrong move:

Skipping the validity check after finding an unknown probability $k$

Why:

Examiners almost always allocate a mark for confirming the distribution is valid

Correct move:

Always state that the sum of probabilities equals 1 and confirm $0 < k < 1$ after solving for the unknown value

Wrong move:

Confusing expected value with the most likely outcome

Why:

Students often assume the expected value is the outcome with the highest probability, which is incorrect

Correct move:

Always calculate expected value using the weighted average formula; it can be a value that is not a possible outcome

Wrong move:

Reversing the variance formula: $(E [X])^{2} - E [X^{2}]$

Why:

This order gives a negative variance, which is impossible for any distribution

Correct move:

Memorize the order: $Var (X) = E [X^{2}] - (E [X])^{2}$ ; if you get a negative result, you reversed the terms

Wrong move:

Forgetting variance scales with the square of a constant for linear transformations

Why:

Students mix up the transformation rules for expected value and variance

Correct move:

Remember: $E [a X + b] = a E [X] + b$ , but $Var (a X + b) = a^{2} Var (X)$

Wrong move:

Assuming all discrete outcomes are equally likely

Why:

Students automatically assign equal probability when it is not stated

Correct move:

Only assume equal probability if the question explicitly states all outcomes are equally likely

5. Quick Reference Cheatsheet

Concept	Core Formula
Validity Check	$0 \leq P (X = x) \leq 1$ , $\sum P (X = x) = 1$
Expected Value	$E [X] = \sum x P (X = x)$
Expected Value of $X^{2}$	$E [X^{2}] = \sum x^{2} P (X = x)$
Variance	$Var (X) = E [X^{2}] - (E [X])^{2}$
Standard Deviation	$σ = Var (X)$
Linear Transformation: Expected Value	$E [a X + b] = a E [X] + b$
Linear Transformation: Variance	$Var (a X + b) = a^{2} Var (X)$

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2025 · 1
Find unknown probability and expected value
2023 · 2
Calculate expected profit and variance
2021 · 1
Variance calculation for discrete distribution

What's Next

The core concepts you learned in this sub-topic are the foundation for all specific discrete probability distributions tested in IB AI SL, most notably the binomial distribution. Every skill from checking validity to calculating expected value and variance applies directly to these specific distributions, so mastery here will make learning those topics much simpler. These concepts also prepare you for continuous probability distributions, which describe outcomes that can take any value within a range, later in the course. Discrete probability questions appear regularly on both Paper 1 and Paper 2, so this sub-topic is a high-yield area for exam preparation.