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.

Examples in context

Example 1. Penalty shootout sequence. A striker scores a penalty with probability $p = 0.8$. In a sequence of $4$ independent attempts, the probability of the specific outcome score-miss-score-score is $0.8 \times 0.2 \times 0.8 \times 0.8 = 0.1024$. The probability of exactly $3$ goals in any order is $\binom{4}{3}(0.8)^3(0.2) = 4 \times 0.512 \times 0.2 = 0.4096$, and the expected number of goals is $np = 4 \times 0.8 = 3.2$.

Example 2. Simulating a survey response. Pollsters model a "yes" response as a Bernoulli trial with $p = 0.45$. Using a calculator, each trial draws a random number in $[0, 1)$ and counts "yes" if it is below $0.45$. Over $200$ simulated respondents the observed proportion $\hat{p}$ clusters near $0.45$ with spread about $\sqrt{\tfrac{0.45 \times 0.55}{200}} \approx 0.035$, so most simulation runs land between $0.38$ and $0.52$.

Try this

Q1. A fair die is rolled; "success" is rolling a $6$. In $3$ rolls, find the probability of exactly one $6$. [3 marks]

• Cue. $\binom{3}{1}\left(\tfrac16\right)\left(\tfrac56\right)^2 = 3 \times \tfrac16 \times \tfrac{25}{36} = \tfrac{75}{216} \approx 0.347$.

Q2. A seed germinates with probability $0.9$. Find the expected number that germinate from $20$ seeds. [2 marks]

• Cue. $np = 20 \times 0.9 = 18$.

Q3. Explain how observed relative frequency relates to theoretical probability as the number of trials increases. [2 marks]

• Cue. By the law of large numbers, $\hat p \to p$ as $n$ grows; small samples can deviate, large samples converge.