# Discrete probability distributions

> IB Mathematics: Applications and Interpretation SL · IB AI SL
> Source: https://www.owlsprep.com/study/ib-math-ai-sl-u4-discrete-probability-distributions/

This sub-topic introduces core concepts for modeling outcomes of discrete random processes. You will learn to verify valid distributions, calculate expected value and variance, and apply these concepts to practical problems common on IB AI SL exams.

**Prerequisites:** [Basic probability rules](https://www.owlsprep.com/study/ib-math-ai-sl-u3-basic-probability-concepts/); [Descriptive statistics for discrete data](https://www.owlsprep.com/study/ib-math-ai-sl-u2-descriptive-statistics/)

## Learning objectives

- Identify discrete random variables and verify valid discrete probability distributions
- Calculate expected value (mean) for discrete probability distributions
- Calculate variance and standard deviation for discrete probability distributions
- Apply core concepts to real-world probability problems

## Discrete Random Variables and Valid Probability Distributions

**Discrete Probability Distribution** — 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.

*Notation:* P(X=x)

*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.

> **tip**
>
> Validity checks are almost always worth at least one mark in exam questions, never skip this step.

## Expected Value (Mean) of a Discrete Distribution

**Expected Value** — 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.

*Notation:* E[X] \text{ or } \mu

*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. $$E[X] = 1 + 3 + 7 + 6.25 + 3 = 20.25$$

> **Exam tip**
>
> For expected profit problems, remember to include negative values for costs or losses. For example, a net loss of \$5 should be entered as -5 in your calculation.

## Variance and Standard Deviation

**Variance** — 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.

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

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

The most commonly used formula for variance in IB AI SL is $\text{Var}(X) = E[X^2] - (E[X])^2$, 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. $$\sigma = \sqrt{31.1875} \approx 5.58$$

## Common pitfalls

- **Wrong:** Skipping the validity check after finding an unknown probability $k$
  - Why it fails: Examiners almost always allocate a mark for confirming the distribution is valid
  - Correct: Always state that the sum of probabilities equals 1 and confirm $0 < k < 1$ after solving for the unknown value
- **Wrong:** Confusing expected value with the most likely outcome
  - Why it fails: Students often assume the expected value is the outcome with the highest probability, which is incorrect
  - Correct: Always calculate expected value using the weighted average formula; it can be a value that is not a possible outcome
- **Wrong:** Reversing the variance formula: $(E[X])^2 - E[X^2]$
  - Why it fails: This order gives a negative variance, which is impossible for any distribution
  - Correct: Memorize the order: $\text{Var}(X) = E[X^2] - (E[X])^2$; if you get a negative result, you reversed the terms
- **Wrong:** Forgetting variance scales with the square of a constant for linear transformations
  - Why it fails: Students mix up the transformation rules for expected value and variance
  - Correct: Remember: $E[aX + b] = aE[X] + b$, but $\text{Var}(aX + b) = a^2 \text{Var}(X)$
- **Wrong:** Assuming all discrete outcomes are equally likely
  - Why it fails: Students automatically assign equal probability when it is not stated
  - Correct: Only assume equal probability if the question explicitly states all outcomes are equally likely

## 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 | $\text{Var}(X) = E[X^2] - (E[X])^2$ |
| Standard Deviation | $\sigma = \sqrt{\text{Var}(X)}$ |
| Linear Transformation: Expected Value | $E[aX + b] = aE[X] + b$ |
| Linear Transformation: Variance | $\text{Var}(aX + b) = a^2 \text{Var}(X)$ |

## 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.

- [Binomial Distribution](https://www.owlsprep.com/study/ib-math-ai-sl-u4-binomial-distribution/)
- [Normal distribution and applications](https://www.owlsprep.com/study/ib-math-ai-sl-u4-normal-distribution-and-applications/)
- [Calculus](https://www.owlsprep.com/study/ib-math-ai-sl-u5-overview/)

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