Bernoulli trials and sequences of Bernoulli trials, sample data analysis (mean, median, mode, range), simulation of random processes, and the relationship between theoretical probability and observed relative frequency

What this dot point is asking

VCAA wants you to recognise Bernoulli trials, compute the probability of sequences of Bernoulli outcomes, analyse sample data using summary statistics, and use simulation to approximate theoretical probabilities. The dot point is the bridge between Unit 1 probability and Unit 3 discrete random variables.

Bernoulli trials

A Bernoulli trial has exactly two outcomes: success (probability $p$) and failure (probability $1 - p$).

Examples:

• A coin flip: heads (success) or tails (failure).
• A multiple-choice question answered randomly: correct or incorrect.
• A free throw in basketball: in or out.
• A manufacturing item: defective or not defective.

Sequences of Bernoulli trials

A Bernoulli sequence is $n$ independent Bernoulli trials with the same success probability $p$.

For a specific sequence of outcomes (e.g., SSFSF for two successes then a failure then a success then a failure):

For "exactly $k$ successes in $n$ trials" (any order):

This is the binomial probability formula, formalised in Unit 3.

Sample data analysis

When repeated trials are conducted, the resulting data can be summarised:

Mean. Sum of values divided by the number of values. Measures the centre.

Median. The middle value when data are ordered. Robust to outliers.

Mode. The most frequent value.

Range. Maximum minus minimum.

Standard deviation. Measures the spread (Unit 3 / 4 will formalise).

Theoretical probability vs observed relative frequency

For a Bernoulli trial with $P(\text{S}) = p$:

• Theoretical probability. $p$ as defined.
• Observed relative frequency. Number of successes in $n$ trials, divided by $n$.

Law of large numbers (informal). As $n$ grows, the observed relative frequency tends to the theoretical probability.

For small $n$, observed frequencies can differ substantially from $p$. For large $n$, the agreement is close.

Simulation

A simulation of a random process uses random numbers (from a calculator or coin/dice) to approximate theoretical probabilities by repeated trials.

Procedure

1. Define the model. What is being simulated (a Bernoulli trial with $p$ specified)?
2. Choose a random source. Calculator RAND, coin, dice.
3. Decide the success condition. "If RAND $< p$, count as success."
4. Run $n$ trials. Record successes.
5. Compute the observed relative frequency. $\hat p =$ (successes) $/ n$.
6. Compare to theoretical. For large $n$, $\hat p \approx p$.

Worked example. Simulating a free throw

A basketball player has free-throw probability $p = 0.7$. Simulate 100 trials.

In each trial: generate RAND. If RAND $< 0.7$, count as a make.

After 100 trials, the observed make rate should be close to 0.70. (It would be approximately normally distributed around 0.70 with standard deviation $\sqrt{0.7 \times 0.3 / 100} = 0.046$, so most observed rates would be in the range 0.61 to 0.79.)

Why simulation matters

Simulation is the practical approach when:

• The theoretical probability is hard to compute.
• The system has many variables interacting.
• You want to estimate the probability empirically without analytical work.

Modern statistical practice uses simulation heavily (Monte Carlo methods, bootstrap inference). VCE Methods introduces the concept at Year 11 level.

Connection to Unit 3 and Unit 4

Unit 3 will formalise the binomial distribution and its mean, variance and standard deviation. Unit 4 will introduce continuous random variables, the normal distribution, sample proportions and confidence intervals.

The Unit 2 foundation is the Bernoulli trial structure, the binomial probability formula (without the formal label), and the empirical vs theoretical distinction.

Common errors

Treating non-independent trials as independent. Bernoulli sequences assume each trial is independent. Drawing without replacement, where the sample changes, is not a Bernoulli sequence.

Confusing $p$ and $1 - p$. $p$ is the success probability; $1 - p$ is the failure probability. Both appear in the binomial formula.

Missing the binomial coefficient. $P(\text{exactly } k \text{ successes}) = \binom{n}{k} p^k (1-p)^{n-k}$. The $\binom{n}{k}$ counts arrangements; without it you have only one specific sequence.

Confusing observed and theoretical. $p$ is fixed (theoretical). $\hat p$ is observed (from data) and varies sample to sample.

Simulation with too few trials. Small $n$ gives unreliable estimates. The agreement improves as $\sqrt{n}$.

In one sentence

A Bernoulli trial has two outcomes (success with probability $p$, failure with $1 - p$), and a sequence of $n$ independent Bernoulli trials produces the binomial probability $P(\text{exactly } k \text{ successes}) = \binom{n}{k} p^k (1-p)^{n-k}$ with expected value $np$; simulation (using random numbers in repeated trials) approximates theoretical probabilities, with the observed relative frequency converging to $p$ as the number of trials grows.