123i - Predicting Relative Frequencies

Lesson 123 — Predicting Relative Frequencies for Related Events

Strand: Probability | Descriptor: AC9M7P01 | Duration: 45 minutes

Learning Intentions

• To predict how often an event will occur in a given number of trials.
• To calculate and interpret relative frequency.

Success Criteria

I can:

1. Predict the expected frequency of an event over trials.
2. Calculate relative frequency from experimental results.
3. Compare an observed relative frequency with the predicted probability.
4. Explain why the two rarely match exactly.

Warmup

(6 minutes — predict, mini whiteboards)

1. A coin is tossed times. About how many heads?
2. A die is rolled times. About how many sixes?
3. A die is rolled times. About how many even numbers?
4. A bag has red and blue counters; one is drawn and replaced times. About how many reds?
5. In Q1, would you be surprised by exactly heads? By ? By ?

Answers: 1. About ; 2. About ; 3. About ; 4. About ; 5. Exactly would be unremarkable but not guaranteed; is entirely ordinary; would be astonishing and would make you suspect the coin.

Q5 is the lesson’s whole idea in miniature. Prediction gives an expectation, not a promise. Results near it are normal; results far from it are evidence something is wrong.

Activities

Activity 1 — Explicit Instruction: Expected Frequency (14 min)

The formula:

I do — the die. , rolled times:

Say what this means precisely: “If we roll a fair die times, we expect about sixes.” Not exactly — about . Getting or would be entirely unremarkable.

We do — predict:

1. A coin tossed times: expected heads?
2. A spinner with equal sectors, spun times: expected landings on any one sector?
3. A bag of red and blue, drawn with replacement times: expected reds?
4. A die rolled times: expected numbers greater than ?

(Answers: ; ; ; , so .)

Relative frequency — the other direction. After an experiment, we can compute what actually happened:

I do. A coin tossed times gives heads:

Predicted probability: . Observed: . Close, not equal — and that is expected.

The vocabulary distinction, drawn on the board:

Term	Comes from	Symbol/example
Probability	Theory — counting outcomes
Relative frequency	Experiment — counting results

The key relationship: as the number of trials grows, relative frequency tends to settle near the theoretical probability. This is why casinos and insurance companies profit reliably from events that are individually unpredictable — Lesson 126 tests it experimentally.

Activity 2 — Predict and Compute Circuit (14 min)

Pairs.

Set A — expected frequency.

1. A die rolled times: expected s? Expected odd numbers?
2. A coin tossed times: expected tails?
3. A spinner of equal sectors spun times: expected landings on sector ? On an even-numbered sector?
4. A bag of green and white, drawn with replacement times: expected greens?
5. A pack of cards, one drawn and replaced times: expected hearts? Expected kings?

Set B — relative frequency.

6. A die rolled times gives sixes. Find the relative frequency and compare with the theoretical probability.
7. A coin tossed times gives heads. Relative frequency? Compare.
8. A spinner spun times lands on red times. Relative frequency as a decimal (3 d.p.)?

Set C — working backwards.

9. A spinner is spun times and lands on blue times. Estimate .
10. A drawing pin is dropped times and lands point-up times. Estimate .
11. Why can Q10’s probability only be estimated, while a die’s can be calculated exactly?

Socratic scaffolding for Set C Q11:

Prompt	Purpose
For a die, how do you find without rolling?	Count outcomes — six faces, equally likely by symmetry.
Is a drawing pin symmetric?	No — there is no reason to expect point-up and point-down to be equally likely.
So how can we find its probability?	Only by experiment: drop it many times and use the relative frequency.
Is that estimate exact?	No — but it improves with more trials.
Name the two approaches.	Theoretical probability (from symmetry) and experimental probability (from data).
When is each available?	Theoretical when outcomes are equally likely by design; experimental always.

(Answers: 1. ; ; 2. ; 3. ; ; 4. ; 5. ; ; 6. vs — a little high; 7. vs — a little low; 8. ; 9. ; 10. ; 11. as scaffolded.)

Activity 3 — Inquiry: how Far off is Too Far? (9 min)

Pairs.

A coin is tossed times. You expect about heads.

1. Would you be suspicious of heads? ? ? ?
2. Where would you draw the line, and why is it hard to say exactly?
3. Now suppose the coin is tossed only times and gives heads. Is that stronger or weaker evidence of an unfair coin than heads in ?

Socratic scaffolding:

Prompt	Purpose
Q1: which results feel normal?	and certainly; raises an eyebrow; is very hard to explain by chance.
Why is the line hard to draw?	Any result is possible with a fair coin — even heads. We are judging plausibility, not possibility.
Q3: which is stronger evidence?	in . Getting in happens reasonably often by chance; in almost never does.
What does that tell you about trial numbers?	The same proportion is far more convincing from more trials.
Looking back	A small sample can be extreme by luck. This is Lesson 120’s point — more data reduces chance variation.

The closing point: deciding when evidence is strong enough to reject “fair” is a real statistical question, and it is what a whole branch of statistics exists to answer. Year 7’s job is to recognise that the question is genuine and that trial count matters.

Checks for Understanding

(5 minutes — exit ticket)

1. A die is rolled times. How many s would you expect?
2. A spinner with equal sectors is spun times. Expected landings on one sector?
3. A coin tossed times gives heads. Find the relative frequency and compare with the predicted probability.
4. A pin dropped times lands point-up times. Estimate .
5. Reasoning. A die rolled times gives sixes instead of the expected . Should you conclude the die is loaded? Explain.

Answers: 1. ; 2. ; 3. , slightly below the predicted — an unremarkable difference; 4. ; 5. Not on this evidence alone. Getting instead of in rolls is well within ordinary variation. More trials would be needed before suspecting the die.

Common Misconceptions

Misconception	How to pre-empt it
Expecting the exact predicted frequency.	”About ”, never “exactly ” — stated at every prediction.
Believing a run of results changes future probability.	Addressed head-on in Lesson 129; flag it here if it arises.
Confusing probability with relative frequency.	The two-row vocabulary table.
Thinking any deviation proves unfairness.	The inquiry’s Q1 and exit Q5.
Believing a proportion from trials is as convincing as from .	Inquiry Q3.
Assuming every situation has a theoretical probability.	The drawing pin — some probabilities can only be estimated.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A spinner has sectors, two of which are red. In spins, how many reds would you expect?

Answer

.

E2 (AMC Junior style). A die is rolled times and gives fives. Find the relative frequency and say whether it is close to expected.

Answer

against the expected . Somewhat high — the expected count was and occurred. Worth more trials before concluding anything.

E3 (Challenge). A bag holds red and blue counters. In draws with replacement, red appears times. Estimate the fraction of red counters, and suggest the simplest bag composition consistent with it.

Answer

Relative frequency , close to . The simplest bag would be red and blue — though red and blue, or any ratio, fits equally.

E4 (Challenge). Two students test the same spinner. One spins times and gets for red; the other spins times and gets . Whose estimate would you trust, and why?

Answer

The second. More trials reduce the influence of chance, so its relative frequency is likely closer to the true probability. Combining both (-ish overall) would be better still.

E5 (Challenge). Why does an insurance company, which cannot predict whether any individual will make a claim, still predict its total claims accurately?

Answer

Individual events are unpredictable, but across hundreds of thousands of customers the relative frequency of claims settles very close to the underlying probability. Large numbers turn individual uncertainty into collective predictability — the same principle as the coin settling near .

Homework

1. Predict the expected frequency: (a) a die rolled times — how many s? (b) a coin tossed times — how many tails? (c) a -sector spinner spun times — how many landings on one sector? (d) a bag of red and blue, drawn times with replacement — how many reds?
2. A die rolled times: expected number of even results? Of results greater than ?
3. Calculate the relative frequency: (a) heads in tosses (b) sixes in rolls (c) reds in spins.
4. For each in Q3, compare with the theoretical probability (assuming fairness) and comment.
5. A drawing pin dropped times lands point-up times. (a) Estimate . (b) Explain why this cannot be calculated theoretically.
6. A spinner is spun times, landing on green times. Estimate and suggest how many of equal sectors might be green.
7. A coin tossed times gives heads. A second coin tossed times gives heads. Which gives stronger evidence of bias? Explain.
8. Reasoning. Explain the difference between probability and relative frequency.
9. Reasoning. Explain why “expected frequency” does not mean the result you will definitely get.
10. Challenge. A game claims . In plays you win times. Calculate the relative frequency, compare with the claim, and say what further evidence you would want.

Answers: Q1 — (a) (b) (c) (d) . Q2 — ; . Q3 — (a) (b) (c) . Q4 — (a) close to ✓ (b) slightly below , unremarkable (c) depends on the spinner’s design; if it has four equal sectors, is somewhat above . Q5 — (a) (b) the pin is not symmetric, so there is no equally-likely argument; only experiment can estimate it. Q6 — , consistent with of sectors being green. Q7 — the second: the same proportion () from ten times as many trials is far less likely to arise by chance. Q8 — probability is calculated from theory by counting equally likely outcomes; relative frequency is measured from actual experimental results. Q9 — it is an average expectation across many repetitions; any single run varies around it by chance. Q10 — against a claimed ; the expected wins were , so is well short. Worth more plays before concluding, but this is enough to warrant scepticism.