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Relative Frequency & Probability · L3 of 4 ~40 min MST-12-S2-09 ⚡ +85 XP available

Relative Frequency and Expected Frequency

Theoretical probability tells you what should happen in the long run. Relative frequency tells you what actually happened in an experiment. As the number of trials grows, these two numbers converge, and that convergence is a powerful bridge between theory and real data.

Today's hook, A fair coin is flipped 10 times and lands heads 7 times. Does this mean P(heads) = 0.7? And if the same coin is flipped 10 000 times with 5 037 heads, which result is more trustworthy? Think about what the difference actually means.

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Recall, your gut answer first

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A fair coin is flipped 10 times and lands heads 7 times. Write your gut answers, no calculating yet:

Does this mean $P(\text{heads}) = 0.7$?
If you flipped the coin 10 000 times and got heads 5 037 times, what is $P(\text{heads})$ now?
What's the difference between what these two experiments tell you?

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The key formulas you need to own

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Relative frequency is a practical estimate of probability built from real data. Expected frequency is the reverse, given a theoretical probability, how many times do you expect the event in $n$ trials? The larger the sample, the more reliable both tools become.

Relative frequency $= f/n$ where $f$ = number of times the event occurred, $n$ = total number of trials. Expected frequency $E = n \times p$ where $n$ = number of trials, $p$ = probability of the event. As $n \to \infty$, relative frequency $\to$ theoretical probability $P(E)$, this is the law of large numbers.

$\text{rel. freq.} = f/n$ | $E = n \times p$ | As $n \uparrow$, rel. freq. $\to P(E)$

Relative frequency ≠ exact probability

Relative frequency is an ESTIMATE. Short experiments produce unreliable estimates; long experiments produce reliable ones. Never claim $P = 7/10$ from just 10 trials as if it were exact.

Expected frequency is not a guarantee

$E = n \times p$ gives an average expectation, not a fixed count. In 100 die rolls, you expect 16–17 sixes, but you might get 12 or 22.

Units check for E

$n$ is a count (dimensionless), $p$ is a probability (between 0 and 1), so $E = n \times p$ is also a count, it can be a decimal (e.g. $E = 33.3$ means "about 33 times in 100 trials").

What you'll master

Know

Key facts

Relative frequency formula: $f/n$
Expected frequency formula: $E = n \times p$
Law of large numbers: as $n$ increases, relative frequency approaches theoretical probability

Understand

Concepts

Why short experiments give unreliable probability estimates
Why we can never say an experiment "proves" a theoretical probability from a finite sample
The practical value of $E = n \times p$ in real-world applications (quality control, medicine, insurance)

Can do

Skills

Calculate relative frequency from experimental data
State whether a relative frequency estimate is reliable and why
Calculate expected frequency given $n$ and $p$
Interpret $E$ in a real-world context

Key terms

Relative frequencyThe ratio $f/n$ of the number of times an event occurred to the total number of trials. Used as a practical estimate of probability from experimental data.

Theoretical probabilityThe mathematically expected probability of an event, calculated from equally likely outcomes: $P(E) = n(E)/n(S)$. What "should" happen in the long run.

Experimental probabilityAnother name for relative frequency, the probability estimate obtained from running an actual experiment rather than calculating theoretically.

TrialsEach individual repetition of a chance experiment. The total number of trials is $n$.

Law of large numbersAs the number of trials $n$ increases, the relative frequency of an event approaches the theoretical probability $P(E)$.

Expected frequency$E = n \times p$, the average number of times an event is expected to occur in $n$ trials, given probability $p$ per trial. Can be a decimal.

Experimental Probability and Relative Frequency

Build the first core idea, then test it with a quick check.

Experimental Probability and Relative Frequency

core concept

When we can't list all outcomes and assign equal probabilities, for example with a biased coin or a complex real-world event, we estimate probability from experimental data. The result is called the relative frequency.

Relative frequency $= f/n$: divide the number of times the event occurred ($f$) by the total number of trials ($n$).

The result is always between 0 and 1, it IS a probability estimate.
With few trials, relative frequency fluctuates widely. With many trials, it stabilises near the theoretical value.
Relative frequency is also called experimental probability.

Biased coin example: A coin is flipped 50 times, landing heads 32 times. Relative frequency of heads $= 32/50 = 0.64 = 64\%$. This is our best estimate of $P(\text{heads})$ based on this experiment. Is the coin fair? 64% is well above 50%, suggesting possible bias, but 50 trials is not enough to be certain.

When to use relative frequency: Use relative frequency when: (1) the theoretical probability isn't known (e.g. probability a patient recovers), (2) you're verifying whether a device is fair, or (3) designing a simulation or model. It's also the probability you use when analysing past data (historical frequency).

Experimental (relative) frequency = number of times event occurred ÷ total number of trials. It is an estimate of probability based on data. As trial count increases, experimental frequency approaches theoretical probability.

Pause, copy the relative frequency formula (experimental probability = number of occurrences ÷ total trials) and note it is an estimate of the true probability, the estimate becomes more accurate as the number of trials increases into your book.

Quick check: A die is rolled 60 times. The number 3 appears 14 times. What is the relative frequency of rolling a 3?

Law of Large Numbers

Connect the next method to the foundation and check your reasoning.

Law of Large Numbers

core concept

Experimental (relative) frequency = occurrences ÷ total trials is an estimate of probability based on real data. This estimate is imprecise for small trial counts but the law of large numbers guarantees it converges toward the theoretical probability as the number of trials increases without limit, not in any finite trial, but in the long run.

One of the most important results in probability is that relative frequencies converge to theoretical probabilities as the number of trials increases. This is called the law of large numbers.

The idea: Run an experiment many times. Plot the relative frequency after each trial. It will jump around at first, then settle closer and closer to the theoretical probability $P(E)$.

Implication: Small samples can mislead. A coin landing 8 heads in 10 flips does not mean the coin is biased, this has a reasonable probability of happening by chance.
Practical cutoff: There's no exact number, but in most HSC contexts, relative frequency from at least a few hundred trials is considered a reasonable estimate of probability.

What "converges" means: The relative frequency doesn't necessarily equal $P(E)$ after $n$ trials. It just gets closer on average. You can still get unlucky streaks. But as $n$ grows very large, the chance of being far from $P(E)$ becomes tiny.

Common misconception, "it's due": If a fair coin lands tails 5 times in a row, the probability of heads on the next flip is still 0.5. Each flip is independent. The law of large numbers is about long-run averages, NOT about individual results "balancing out" in the short run.

Law of large numbers: as the number of trials increases, the experimental (relative) frequency of an event gets closer and closer to the theoretical probability. It does not guarantee a result in any finite number of trials.

Pause, copy the law of large numbers: as the number of trials increases, the relative frequency of an event approaches its theoretical probability, and note it applies to the long run, not any specific finite sample: short-run results can still vary widely into your book.

True or false: If a fair coin lands heads 7 times in the first 10 flips, it is more likely to land tails in the next flip because "it needs to balance out."

Expected Frequency

Apply the third core idea and confirm the key calculation.

Expected Frequency

core concept

The law of large numbers says relative frequency converges to theoretical probability as n grows. Working in the forward direction: if you know the probability and the number of trials, expected frequency = P × n gives the average number of times the event should occur. For example, P(6) = 1/6 on a die, so in 120 rolls the expected frequency of a 6 is (1/6) × 120 = 20.

If you know the theoretical probability of an event and you're about to run an experiment, you can calculate how many times you expect the event to occur: $E = n \times p$.

Formula: $E = n \times p$, where $n$ = number of trials, $p$ = probability per trial.

Interpretation: $E$ is the average number of times you expect the event over many repetitions of the $n$ trials. It's not guaranteed for any single run.
Non-integer $E$: $E$ can be a decimal. $E = 33.3$ means "on average, about 33 times per 100 trials."

Quality control example: A machine produces lightbulbs, and 2% are defective ($p = 0.02$). In a batch of 500 bulbs, expected number defective: $E = 500 \times 0.02 = 10$ bulbs. A quality inspector would flag the batch if the actual count is significantly more than 10.

Insurance and medicine: Insurance companies use $E = n \times p$ to set premiums. If $P(\text{claim}) = 0.05$ and there are 10 000 policyholders, $E = 10\,000 \times 0.05 = 500$ claims expected. Medical researchers use it to predict trial outcomes.

Expected frequency = theoretical probability × number of trials. E.g., P(6) = 1/6 on a die; in 120 rolls, expected frequency of 6 = (1/6) × 120 = 20. Expected frequency is a prediction; actual frequency will vary.

Pause, copy the expected frequency formula: expected frequency = P × n, with one worked example (e.g., P(head) = 0.5, n = 80 flips → expected 40 heads), and note it is a long-run prediction, actual frequency in any experiment will vary into your book.

Fill the blanks: A spinner has $P(\text{blue}) = 0.25$. In 200 spins, the expected number of blue outcomes is $E = 200 \times$ $=$ . If the spinner lands on blue 38 times, the relative frequency is .

Work through the examples

Follow the worked solutions, then transfer the method to the final quick tasks.

Worked examples · 3 problems

PROBLEM 1 · RELATIVE FREQUENCY FROM DATA

A die is rolled 120 times. The results are: 1→22, 2→18, 3→24, 4→19, 5→21, 6→16. (a) Find the relative frequency of each outcome. (b) Compare with the theoretical probability $P(\text{any face}) = 1/6 \approx 0.167$. (c) Does the data suggest the die is biased? Explain.

Part (a), relative frequencies
$1: 22/120 = 11/60 \approx 0.183$
$2: 18/120 = 3/20 = 0.150$
$3: 24/120 = 1/5 = 0.200$
$4: 19/120 \approx 0.158$
$5: 21/120 = 7/40 = 0.175$
$6: 16/120 = 2/15 \approx 0.133$

Divide each frequency by the total number of trials (120). Check: all six relative frequencies should sum to 1. $22+18+24+19+21+16 = 120$ ✓

PROBLEM 2 · EXPECTED FREQUENCY

In a large city, 8% of adults have a particular genetic marker. (a) In a sample of 350 adults, how many would be expected to have the marker? (b) If 40 adults in the sample have the marker, is this unusual? (c) What relative frequency does 40 out of 350 represent?

Part (a), expected frequency
$p = 0.08$, $n = 350$
$E = n \times p = 350 \times 0.08 = 28$ adults

Convert 8% to a decimal: $p = 0.08$. Multiply by the sample size to get the expected count.

PROBLEM 3 · COMBINING RELATIVE FREQUENCY AND EXPECTED FREQUENCY

A factory quality-control team tested 500 items and found 15 defective. (a) Estimate the probability of a defective item using relative frequency. (b) In the next production run of 2 000 items, how many defective items are expected? (c) The manager wants fewer than 40 defects in 2 000 items. Is this target reasonable based on the current data?

Part (a), estimate $P(\text{defective})$
$\text{rel. freq.} = 15/500 = 3/100 = 0.03 = 3\%$
So estimated $P(\text{defective}) \approx 0.03$

Relative frequency from the 500-item test is used as the best available estimate of probability, since the true theoretical probability is unknown.

Match each statement about the law of large numbers with TRUE or FALSE:

Relative frequency always equals theoretical probability after 100 trials
A fair coin landing heads 6 times in 10 flips means P(heads) = 0.6
As the number of trials increases, relative frequency approaches theoretical probability
Expected frequency E = n × p is an average, not a guaranteed count

True, this is the law of large numbers
True, it is a long-run average
False, no number of trials guarantees it
False, it is an unreliable estimate

Top 3 list: Name THREE real-world situations where expected frequency $E = n \times p$ would be used to make a decision.

Show what you can do

Answer the exam-style questions, then compare your working with the model answers.

Short answer, exam-style questions

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ApplyBand 33 marks

SA 1. A spinner is spun 80 times. It lands on green 28 times. (a) Calculate the relative frequency of green. (b) If the theoretical probability of green is 0.30, how does the relative frequency compare? (c) How many times would you expect green in 500 spins? (3 marks)

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ApplyBand 3–43 marks

SA 2. In a raffle with 1 200 tickets, $P(\text{winning}) = 1/400$. (a) How many winning tickets are there? (b) If 3 600 tickets are sold across three identical raffles, what is the expected number of winners? (c) Explain why this expected frequency does not guarantee that exactly this many winners will occur. (3 marks)

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AnalyseBand 4–54 marks

SA 3. A medical study tests a drug on 800 patients. The drug is effective for 680 patients. (a) Calculate the relative frequency of effectiveness. (b) Use this as an estimate of the probability the drug works. (c) In a hospital that plans to prescribe the drug to 250 patients, calculate the expected number for whom the drug will be effective. (d) A doctor says "the drug works 85% of the time." Is this consistent with the study data? Justify. (4 marks)

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📖 Comprehensive answers (click to reveal)

SA 1 (3 marks): (a) $28/80 = 7/20 = 0.35 = 35\%$ [1]. (b) 0.35 is above the theoretical 0.30, slightly higher than expected, but with only 80 spins this variation is plausible [1]. (c) $E = 500 \times 0.30 = 150$ [1].

SA 2 (3 marks): (a) $1200 \times 1/400 = 3$ winning tickets [1]. (b) $E = 3600 \times 1/400 = 9$ winners [1]. (c) Expected frequency is a long-run average, not a guaranteed count for one set of raffles, so the actual number of winners can vary from 9 [1].

SA 3 (4 marks): (a) $680/800 = 17/20 = 0.85 = 85\%$ [1]. (b) $P(\text{effective}) \approx 0.85$ [1]. (c) $E = 250 \times 0.85 = 212.5 \approx 213$ patients [1]. (d) The claim "85% of the time" is exactly consistent with the study data (relative frequency $= 0.85$). The doctor is using the study's relative frequency as an estimate of theoretical probability [1].

Retrieve, reflect and finish

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Revisit your thinking

Coin: 10 flips, 7 heads → relative frequency $= 7/10 = 0.7$. This is an ESTIMATE of $P(\text{heads})$, not the exact value. From 10 000 flips with 5 037 heads → relative frequency $= 5037/10000 = 0.5037$. This is much closer to the theoretical $P(\text{heads}) = 0.5$, because more trials produce a more reliable estimate. The difference: 10 flips is too few to draw conclusions; 10 000 flips gives a reliable estimate that is very close to the theoretical probability.

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Relative Frequency & Probability · Maths Standard