Relative Frequency and Probability

This focus area was worth 9 marks in NESA's sample HSC paper for the 2024 syllabus, much more than probability was worth in recent HSC papers under the old syllabus. Expect multiple choice questions on the basics and written questions on tree diagrams, two-way tables and expected frequency.

• Probabilities range from 0 (impossible) to 1 (certain), as fractions, decimals or percentages.
• The sample space and the number of outcomes of an experiment.
• Probability of an event with equally likely outcomes.
• Complementary events: $P(\text{not }A)=1-P(A)$
• Multistage events using arrays, tables and tree diagrams, and $P(A\text{ and }B)=P(A)\times P(B)$ when $A$ and $B$ are independent.
• Relative frequency as an estimate of probability, and expected frequency $np$.
• Two-way tables and Venn diagrams for two attributes.
• How statistics and probabilities are used by the media, governments and companies.

An experiment (or trial) is something with an uncertain result, like rolling a die. Each possible result is an outcome, and the set of all outcomes is the sample space. An event is one or more outcomes, such as "roll an even number".

If all outcomes are equally likely:

For a fair die, $P(\text{even})=\frac{3}{6}=\frac{1}{2}$ which is 0.5 or 50%.

Complementary events

The complement of an event is "the event does not happen". The two probabilities add to 1:

If the probability a flight is on time is 0.82, the probability it is not on time is $1-0.82=0.18$. The complement is the quickest way to find "at least one": $P(\text{at least one})=1-P(\text{none})$

A multistage experiment has two or more steps, such as rolling two dice or drawing two cards.

Arrays (tables of outcomes)

For two steps with equally likely outcomes, list every combination in a grid. Rolling two dice gives $6\times 6=36$ outcomes. The sum is 8 for (2, 6), (3, 5), (4, 4), (5, 3) and (6, 2), so $P(\text{sum}=8)=\frac{5}{36}$

Tree diagrams

A tree diagram shows each stage as a set of branches, with the probability on each branch.

• The branches from one point add to 1.
• Multiply along a path to find the probability of that path: $P(A\text{ and }B)=P(A)\times P(B)$ when $A$ and $B$ are independent; for dependent stages, use the changed probability on the second branch.
• Add the probabilities of all paths that give the event you want.

Worked example: two free throws

A basketballer scores each free throw with probability 0.7, and the two throws are independent. Find the probability that she scores (a) both, (b) exactly one, (c) at least one.

The first throw has branches score (0.7) and miss (0.3); from each, the second throw has the same two branches. There are four paths.

(a) Both scored:

(b) Exactly one: two paths, score then miss, or miss then score.

(c) At least one: use the complement of missing both.

Check: the four paths add to $0.49+0.21+0.21+0.09=1$

Without replacement

When the second stage depends on the first, such as choosing two people from a group, the branch probabilities change after the first choice. From 4 girls and 6 boys, choosing two at random:

After one girl is chosen, there are 3 girls left out of 9 people.

Relative frequency

When outcomes aren't equally likely (a drawing pin landing point up, a machine making a faulty part), estimate the probability from an experiment or past data:

where $f$ is the number of times the event occurs and $n$ is the number of trials.

If a drawing pin lands point up 315 times in 500 throws, the relative frequency is $\frac{315}{500}=0.63$. The more trials, the closer the relative frequency tends to get to the true probability. A small number of trials can give a misleading estimate.

Expected frequency

The expected frequency of an event in $n$ trials is

where $p$ is the probability of the event in one trial. For a fair die rolled 150 times, a 5 or 6 is expected $150\times \frac{1}{3}=50$ times. Using the drawing pin estimate, in 2000 throws we'd expect $2000\times 0.63=1260$ to land point up. Expected frequency is a long-run average, not a guarantee.

Two-way tables

A two-way table sorts people or items by two attributes at once, with totals for each row and column.

Worked example: paying at a café

A café recorded how 200 customers paid.

(a) A customer is chosen at random. Find the probability they paid cash.

(b) A customer aged 30 or over is chosen. Find the probability they paid cash.

The denominator changes to 120 because only customers aged 30 and over are being considered. Read the question carefully for words like "of those who…" or "given that…".

(c) Out of 1500 customers next month, how many would you expect to pay by card?

About 1215 customers.

Venn diagrams

A Venn diagram shows two overlapping groups inside a rectangle (everyone). Fill in the overlap first, then each "only" region, then the outside.

Example. Of 60 students, 28 study music, 19 study art and 7 study both. Music only is $28-7=21$, art only is $19-7=12$, and neither is $60-21-12-7=20$. So $P(\text{neither})=\frac{20}{60}=\frac{1}{3}$

Governments use statistics and probabilities to plan for floods and bushfires and to decide on health programs such as screening. Companies use them to price insurance, warranties and games of chance (which are designed so the company wins in the long run). The media report risks and survey results. When you're asked to evaluate a claim, ask: how big was the sample, was it random, and is the probability being compared fairly (a 50% increase in a very small risk is still a small risk)?

• Adding along the branches of a tree diagram instead of multiplying.
• Forgetting there are two paths for "exactly one" (score then miss, and miss then score).
• Not changing the second-stage probabilities when items are not replaced.
• Using the grand total as the denominator when the question restricts the group.
• Counting the overlap of a Venn diagram twice.
• Treating expected frequency as exactly what will happen.

• Give probabilities as fractions in simplest form, or decimals, unless a format is asked for. Don't round fractions like $\frac{5}{36}$ unless told to.
• Draw the full tree diagram, with probabilities on every branch, even if you only need one path; it often earns a mark.
• For "at least one", use $1-P(\text{none})$.
• In two-way table questions, circle the row or column the question restricts you to before you divide.

• Data analysis (Year 11): frequency tables and survey data.
• The normal distribution: probabilities from z-scores and the empirical rule.
• Managing money (Year 11): insurance and the cost of risk.