Discrete random variables and the binomial distribution

Mathematics - Analysis and Approaches · IB Diploma Programme (DP) - SL & HL · 5 min read · Updated: 7 Sept 2026

Introduction to Discrete Random Variables

Welcome to the world of Discrete Random Variables! While basic probability tells us the "chance" of a single event happening, this chapter allows us to look at the "big picture." We will learn how to map out every possible outcome of an experiment and calculate what we should expect to happen in the long run. This is a vital skill for everything from predicting board game wins to calculating insurance risks.

1. What is a Discrete Random Variable?

A random variable (usually written as a capital letter like $X$) is a quantity whose value depends on the outcome of a random event.
The word discrete means the variable can only take specific, separate values (like 0, 1, 2...) that you can count. You can't have "2.4" heads when tossing a coin!

The Probability Distribution

A probability distribution is simply a list or a table that shows all possible values of $X$ and the probability of each one occurring. For any valid probability distribution, two rules must be true:

1. Every individual probability must be between 0 and 1: $0 \le P(X=x) \le 1$.
2. The sum of all probabilities must equal exactly 1: $\sum P(X=x) = 1$.

Common Exam Trick: You might be given a table with a missing value (like $k$). Just add up all the given probabilities and subtract from 1 to find $k$!

2. Expected Value (Mean)

The Expected Value, written as $E(X)$ or $\mu$, is the average value you would expect to get if you repeated the experiment many, many times. It is the "long-term average."

The formula is:
$E(X) = \sum x \cdot P(X=x)$

Step-by-Step: How to calculate $E(X)$
1. Take each value of $x$ from your table.
2. Multiply it by its corresponding probability.
3. Add all those results together.

Example: If you have a 20% chance of winning \$10 and an 80% chance of winning \$0, your expected value is $(10 \cdot 0.20) + (0 \cdot 0.80) = \$2$. This doesn't mean you will win \$2 in one game; it means if you play 100 times, you’ll likely walk away with about \$200 total.

Key Takeaway: The sum of probabilities is always 1, and the Expected Value is the weighted average of the outcomes.

3. The Binomial Distribution

The Binomial Distribution is a specific type of discrete probability distribution. We use it when we are looking for the number of "successes" in a fixed number of trials.

When to use Binomial (The "BINS" Mnemonic)

To use the Binomial model, the experiment must meet these four criteria:

B – Binary: There are only two possible outcomes (Success or Failure).
I – Independent: One trial does not affect the next.
N – Number: There is a fixed number of trials ($n$).
S – Same: The probability of success ($p$) is the same for every trial.

Notation and Formulas

If a variable $X$ follows a binomial distribution, we write: $X \sim B(n, p)$
Where $n$ is the number of trials and $p$ is the probability of success.

The probability of getting exactly $r$ successes is:
$P(X = r) = \binom{n}{r} p^r (1-p)^{n-r}$

Don't worry if this formula looks scary! $\binom{n}{r}$ is the "combinations" button on your calculator (nCr), representing the number of ways to arrange the successes. $p^r$ is the probability of your successes, and $(1-p)^{n-r}$ is the probability of your failures.

Mean and Variance of Binomial

For a binomial distribution, the formulas for the mean and variance are very simple:
Mean: $E(X) = np$
Variance: $Var(X) = np(1-p)$

4. Using Technology (GDC)

In Paper 2, you should use your Graphic Display Calculator (GDC) to save time and avoid errors. There are two main functions you need to know:

1. Binomial PDF: Use this for "exactly" questions. (e.g., $P(X = 3)$)
2. Binomial CDF: Use this for "range" questions. (e.g., $P(X \le 3)$)

Common Mistake: If the question asks for $P(X > 3)$, your calculator usually only does "less than or equal to." You must calculate $1 - P(X \le 3)$. Always check if the inequality includes the number (at least, at most, more than, less than)!

Quick Review:
- "Exactly 5" $\implies$ binompdf
- "At most 5" ($0, 1, 2, 3, 4, 5$) $\implies$ binomcdf
- "More than 5" ($6, 7, ...$) $\implies$ $1 -$ binomcdf(up to 5)

5. AHL Only: Variance of Discrete Random Variables

Standard Level students can skip this part, but HL students need to know this for any discrete distribution (not just binomial).

The Variance measures how spread out the values are from the mean. While SL students only need the shortcut for Binomial, HL students must be able to find the variance for any discrete table using this formula:

$Var(X) = E(X^2) - [E(X)]^2$

How to calculate $E(X^2)$:
1. Square each $x$ value in your table.
2. Multiply that squared value by its probability.
3. Sum them up.

How to calculate Variance:
1. Find $E(X^2)$ as shown above.
2. Find $E(X)$ and square it.
3. Subtract the second from the first.

Note: To find the Standard Deviation ($\sigma$), simply take the square root of the variance: $\sigma = \sqrt{Var(X)}$.

Summary and Key Takeaways

1. Discrete Data: You can count it. The sum of all probabilities is always 1.
2. Expected Value: Multiply across the table and add. It's the long-term average.
3. Binomial: Use when you have "Success/Failure" and a fixed number of tries. Remember BINS.
4. Calculator: Use pdf for exact values and cdf for cumulative (up to) values.
5. Common Trap: Always read the wording carefully—"more than 2" means you start counting at 3!

Need more? This chapter leads directly into "The normal distribution," which handles continuous data (measuring instead of counting).

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is a discrete random variable?

A quantity whose value depends on a random event and can only take specific, separate, countable values.

What are the two fundamental rules for a valid probability distribution?

1. Each probability must be between 0 and 1 (inclusive). 2. The sum of all probabilities must equal exactly 1.

How is the Expected Value $E(X)$ of a discrete random variable calculated?

$E(X) = \sum x \cdot P(X=x)$

What are the four conditions required to use a Binomial Distribution (the BINS mnemonic)?

Binary (Success/Failure), Independent trials, fixed Number of trials (n), and the Same probability of success (p).

What is the formula for calculating the probability of exactly $r$ successes in a binomial distribution?

$P(X = r) = \binom{n}{r} p^r (1-p)^{n-r}$

What are the simple formulas for the Mean and Variance of a Binomial Distribution?

$E(X) = np$ and $Var(X) = np(1-p)$

When should you use Binomial PDF versus Binomial CDF on a GDC?

Use PDF for the probability of an exact value ($P(X=r)$); use CDF for a cumulative range ($P(X \le r)$).

How do you calculate $P(X > r)$ using a calculator that only performs $P(X \le r)$?

Use the complement rule: $P(X > r) = 1 - P(X \le r)$.

How is the variance of a general discrete random variable calculated (AHL)?

$Var(X) = E(X^2) - [E(X)]^2$

How do you find the Standard Deviation from the Variance (AHL)?

$\sigma = \sqrt{Var(X)}$

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