QCE Mathematical Methods - Unit 3 - Discrete random variables

Binomial Distributions | QCE Mathematical Methods

Learn QCE Mathematical Methods binomial distributions, including Bernoulli trials, binomial probabilities, mean, variance and wording traps.

Updated 2026-05-18 - 5 min read

A binomial random variable counts how many successes occur in a fixed number of independent Bernoulli trials. If $X$ is binomial, we write:

$ X\sim B(n,p) $

where $n$ is the number of trials and $p$ is the probability of success on each trial.

You can think of a binomial random variable as a sum of $n$ Bernoulli random variables:

$ X=X_1+X_2+\cdots+X_n $

where each $X_i$ is $1$ for success and $0$ for failure. The total $X$ is the number of successes.

Conditions

Before using a binomial model, check:

• fixed number of trials
• each trial has two outcomes
• trials are independent
• probability of success stays the same
• $X$ counts the number of successes

Bernoulli is the special case $n=1$. If $X\sim B(1,p)$, then $X$ has the same distribution as $\operatorname{Bernoulli}(p)$.

Probability formula

The probability of exactly $r$ successes is:

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

The combination $\binom{n}{r}$ counts how many different orders can produce $r$ successes in $n$ trials.

Here is the reason the formula has those parts:

• $p^r$ is the probability of the $r$ successful trials.
• $(1-p)^{n-r}$ is the probability of the remaining failures.
• $\binom{n}{r}$ counts the number of positions where the successes could occur.

For example, with $5$ trials and exactly $2$ successes, one possible order is $SSFFF$. It has probability $p^2(1-p)^3$. But the successes could be in many different positions, and there are $\binom{5}{2}=10$ such positions. That is why the combination multiplier is needed.

Mean and variance

For $X\sim B(n,p)$:

$ E(X)=np $

$ \operatorname{Var}(X)=np(1-p) $

The mean $np$ is the long-run average number of successes per set of $n$ trials.

The formulas also come from the Bernoulli-sum idea. If each trial has mean $p$, then $n$ trials have total mean $np$. If the trials are independent, the variances add, giving $np(1-p)$.

Exact, cumulative and complement probabilities

Methods questions often ask for a range of counts rather than one exact count. Translate the wording before touching technology:

| Wording | Probability statement | Efficient approach | | --- | --- | --- | | exactly $r$ | $P(X=r)$ | use the formula or binomial PDF | | no more than $r$ | $P(X\le r)$ | cumulative probability | | fewer than $r$ | $P(X<r)=P(X\le r-1)$ | adjust the endpoint | | at least $r$ | $P(X\ge r)$ | often $1-P(X\le r-1)$ | | more than $r$ | $P(X>r)$ | often $1-P(X\le r)$ |

Worked example

Worked cumulative example

Wording traps

Binomial questions often hide the calculation inside probability language:

• "at most 3" means $P(X\le 3)$
• "at least 3" means $P(X\ge 3)$
• "more than 3" means $P(X>3)$
• "less than 3" means $P(X<3)$
• "between 2 and 5 inclusive" means $P(2\le X\le 5)$

For technology, distinguish "exactly" from "cumulative". On most calculators, a binomial probability command gives $P(X=r)$, while a binomial cumulative command gives $P(X\le r)$. If you need a middle range such as $P(3\le X\le 7)$, use:

$ P(3\le X\le 7)=P(X\le 7)-P(X\le 2) $

Model suitability

The binomial model is a strong fit when the question can be restated as: "In $n$ independent repeats of the same yes/no experiment, how many successes occur?" If that sentence feels unnatural, pause before using binomial.

Good binomial contexts include repeated guessing with the same number of options, independent free throws with a fixed success rate, repeated quality-control checks where each item has the same defect probability, or a fixed-size random sample where each person is classified as having or not having a characteristic.

Less suitable contexts include drawing cards without replacement from a small deck, waiting until the first success, or counting categories with more than two outcomes unless you clearly combine them into success and failure.

Common mistake

Other situations that break the binomial model include changing success probability, stopping after a certain number of successes, or counting something that is not a number of successes. If the question says "until the first success", that is not binomial because $n$ is not fixed.

Quick check

Sources

• QCAA Mathematical Methods subject page
• QCAA Mathematical Methods 2025 syllabus
• OpenStax Introductory Statistics 2e: Binomial Distribution
• OpenStax Introductory Statistics 2e: Probability Distribution Function for a Discrete Random Variable