Bernoulli trials, the binomial distribution $X \sim \mathrm{Bi}(n, p)$, its probability function $P(X = x) = \binom{n}{x} p^x (1-p)^{n-x}$, mean $E(X) = np$, and variance $\mathrm{Var}(X) = np(1-p)$

What this dot point is asking

VCAA wants you to recognise when a situation is binomial, write down $X \sim \mathrm{Bi}(n, p)$, compute exact binomial probabilities by hand using $\binom{n}{x} p^x (1-p)^{n-x}$ for small $n$ on Paper 1, and use CAS on Paper 2 to compute cumulative probabilities. The mean $np$ and variance $np(1-p)$ are essential shortcuts.

Bernoulli trials

A Bernoulli trial is a single experiment with exactly two outcomes (commonly called "success" and "failure"), where the probability of success is some fixed $p$ (and probability of failure is $1 - p$).

A Bernoulli random variable takes the value 1 for success and 0 for failure, with probability distribution $P(Y = 1) = p$ and $P(Y = 0) = 1 - p$. Its mean is $p$ and its variance is $p(1 - p)$.

The binomial distribution

When $n$ Bernoulli trials are performed under the four binomial conditions, the total number of successes follows a binomial distribution.

The four conditions for a binomial

1. Fixed number of trials $n$.
2. Each trial has only two outcomes, success or failure.
3. The probability of success $p$ is constant across trials.
4. The trials are independent.

If any one of these fails, $X$ is not binomial. The most common failure is sampling without replacement (which breaks independence for finite populations).

Notation: $X \sim \mathrm{Bi}(n, p)$. The two parameters $n$ and $p$ fully determine the distribution.

The binomial probability formula

For $X \sim \mathrm{Bi}(n, p)$ and $x \in \{0, 1, 2, \dots, n\}$:

where $\binom{n}{x} = \frac{n!}{x! (n - x)!}$ counts the number of ways to choose which $x$ of the $n$ trials are successes.

The three factors have natural interpretations:

• IMATH_27 : number of distinct orderings of $x$ successes and $n - x$ failures.
• IMATH_30 : probability of $x$ specific successes occurring.
• IMATH_32 : probability of the remaining $n - x$ trials being failures.

Worked example

A biased coin lands heads with probability $0.6$. It is tossed 5 times. Find the probability of exactly 3 heads.

$X \sim \mathrm{Bi}(5, 0.6)$.

$P(X = 3) = \binom{5}{3} (0.6)^3 (0.4)^2 = 10 \cdot 0.216 \cdot 0.16 = 0.3456$.

Mean and variance

For $X \sim \mathrm{Bi}(n, p)$:

These shortcut formulas come directly from the linearity rule applied to the sum of $n$ Bernoulli trials.

For the biased coin above: $E(X) = 5 \cdot 0.6 = 3$, $\mathrm{Var}(X) = 5 \cdot 0.6 \cdot 0.4 = 1.2$.

Cumulative binomial probabilities

For "at least", "at most" or "between" probabilities, you sum binomial terms.

• IMATH_41
• IMATH_42
• IMATH_43

On Paper 1, $n$ is small enough to compute terms by hand. On Paper 2, use the CAS binomCdf(n, p, lower, upper) function (or its equivalent on your model).

Worked example

A fair die is rolled 10 times. Find the probability of at least 3 sixes.

$X \sim \mathrm{Bi}(10, 1/6)$ where $X$ is the number of sixes.

$P(X \geq 3) = 1 - P(X \leq 2) = 1 - [P(X = 0) + P(X = 1) + P(X = 2)]$.

$P(X = 0) = (5/6)^{10}$.

$P(X = 1) = 10 (1/6)(5/6)^9$.

$P(X = 2) = 45 (1/6)^2 (5/6)^8$.

Sum these (or use CAS) and subtract from 1. The result is approximately $0.2248$.

Worked example: solving for the parameter

A biased coin is tossed 10 times. The probability of getting exactly 7 heads is $0.215$. Find $p$.

$P(X = 7) = \binom{10}{7} p^7 (1 - p)^3 = 120 p^7 (1 - p)^3 = 0.215$.

This is a one-variable equation in $p$. Use CAS solve, restricting to $p \in (0, 1)$: $p \approx 0.6$.

Common Paper 1 traps

Including the constant term twice. The formula has $\binom{n}{x}$ once. Writing $\binom{n}{x} p^x (1 - p)^{n - x}$ but then "rounding up" the binomial coefficient is a sign of confusion.

Forgetting independence. Drawing balls without replacement breaks independence; the resulting distribution is hypergeometric, not binomial.

Wrong formula direction. $E(X) = n p$, not $E(X) = p$ or $E(X) = n$. Likewise variance has both $n$ and $p(1 - p)$.

Boundary errors on cumulative probabilities. "At least 3" means $X \geq 3$, which is $1 - P(X \leq 2)$. Subtracting $P(X \leq 3)$ instead loses the $P(X = 3)$ term.

Squaring the standard deviation incorrectly. $\mathrm{Var}(X) = n p (1 - p)$; do not omit the $(1 - p)$ factor.

In one sentence

The binomial distribution $X \sim \mathrm{Bi}(n, p)$ models the total number of successes in $n$ independent Bernoulli trials with constant success probability $p$, with probability function $P(X = x) = \binom{n}{x} p^x (1 - p)^{n - x}$, mean $np$ and variance $np(1 - p)$; Paper 1 expects by-hand calculations for small $n$ and Paper 2 expects fluent CAS use of binomPdf and binomCdf.