127i - Conducting Repeated Chance Experiments

Lesson 127 — Conducting Repeated Chance Experiments

Strand: Probability | Descriptor: AC9M8P03 | Duration: 45 minutes

Learning Intentions

• To conduct repeated hands-on chance experiments — coins, dice and spinners — following a fair, consistent procedure.
• To record results systematically and calculate relative frequency from experimental data.

Success Criteria

I can:

1. Conduct a repeated chance experiment using a consistent, fair procedure.
2. Record outcomes in a tally and frequency table across many trials.
3. Calculate relative frequency as frequency divided by number of trials.
4. Compare an experimental relative frequency with the theoretical probability and describe the difference.

Warmup

(5 minutes — mini whiteboards, rapid-fire)

1. A fair coin: state the theoretical .
2. A fair six-sided die: state the theoretical .
3. A spinner has equal sectors, of them red. State the theoretical .
4. State the expected frequency formula, in words.
5. If a fair coin is tossed times, how many heads would you expect?

Answers: 1. ; 2. ; 3. ; 4. Expected frequency probability number of trials; 5. .

Activities

Activity 1 — Explicit Instruction: Running and Recording a Trial (12 min)

I do / We do / You do. Materials: coins, dice.

I do: Toss a coin times, tallying as you go.

Outcome	Tally	Frequency
Heads	|||| |||| |
Tails	|||| ||||

Say aloud: “Both are reasonably close to the theoretical , given only trials. A single trial run rarely lands exactly on the theoretical value.”

We do: Together roll a die times (recorded results): , , , , , .

Compare with the theoretical for each face — some are above, some below, but none is wildly off.

You do: In pairs, toss a coin times. Record every toss with a tally, then calculate your own relative frequency of heads and tails.

Activity 2 — Hands-on Experiment: the Three-colour Spinner (14 min)

Pairs. Materials: a spinner (or an improvised paperclip-and-pencil spinner) divided into equal sectors: red, blue, green.

1. State the theoretical probability of each colour.
2. Predict the expected frequency of each colour over spins.
3. Spin times, tallying each outcome.
4. Calculate the relative frequency of each colour (3 d.p.).
5. Compare each relative frequency with the theoretical probability. Which colour was closest? Which was furthest?

Answers: 1. Red , blue , green . 2. Red , blue , green . 3–5. Student’s own data; comparisons should be reported as both an absolute difference (e.g. ” more than expected”) and a relative frequency versus theoretical probability.

Activity 3 — Inquiry: Pooling the Class Data (8 min)

Whole class.

Combine your pair’s coin tosses from Activity 1 with three other pairs’ results, to make a class total of tosses.

1. Recalculate the relative frequency of heads using the pooled tosses.
2. Is the pooled relative frequency closer to than your own pair’s -toss result was?
3. Why might combining more data tend to bring the relative frequency closer to the theoretical probability?

Discussion points: Individual pairs’ results vary more because small samples are more strongly affected by a few unlucky or lucky tosses. Pooling averages out these swings across more trials. (This idea — the law of large numbers — will be studied formally in Lesson 129.)

Checks for Understanding

(6 minutes — exit ticket, collected)

1. Define relative frequency.
2. A die is rolled times, giving six s. Find the relative frequency of rolling a , and compare it with the theoretical probability.
3. Why should a chance experiment’s procedure (e.g. toss height, spin technique) stay the same on every trial?
4. A spinner with equal sectors (only gold) is spun times, giving gold times. Find the relative frequency of gold, and state whether this seems far from the theoretical value.

Answers: 1. The number of times an event occurs divided by the total number of trials. 2. , compared with the theoretical — noticeably lower. 3. Keeping the procedure consistent ensures every trial has the same underlying probabilities; an inconsistent method could bias results, making it unclear whether any variation is due to chance or to the procedure itself. 4. , compared with the theoretical — lower, but with only trials this could still be ordinary chance variation.

Common Misconceptions

Misconception	How to pre-empt it
Expecting a small sample’s relative frequency to exactly equal the theoretical probability.	Contrast several pairs’ Activity 2 results side by side — all differ slightly, and that is expected.
Changing the experimental procedure partway through (e.g. spinning harder, dropping the die from a different height).	State explicitly, before starting, that every trial must use the same physical method.
Confusing frequency (a count) with relative frequency (a proportion).	Always write both the fraction and its simplified/decimal form, labelled clearly.
Believing a run of one outcome makes a different outcome “due” next.	Revisit the gambler’s fallacy from Lesson 121: each trial is independent of previous ones.
Rounding relative frequency too early, before comparing with the theoretical value.	Require at least decimal places before any comparison is made.

Enrichment — Competition-Style Problems

E1 (AMC Junior style). A coin is tossed times, giving heads. Find the relative frequency of heads as a decimal and as a fraction in simplest form.

Answer

Already in simplest form, since .

E2 (Kangaroo style). A spinner has equal sectors, numbered –. It is spun times; the number appears times. Find the expected frequency of , and the difference between observed and expected.

Answer

Three fewer than expected.

E3 (Challenge). In trials, an event’s relative frequency was . How many additional trials — all resulting in the event occurring — would be needed to raise the relative frequency to exactly ?

Answer

Occurrences so far: . Let be the extra trials, all successes:

Check: ✓

Homework

1. Define relative frequency in your own words.
2. A coin is tossed times, giving heads. Find the relative frequency (as a decimal) and compare it with the theoretical probability.
3. A die is rolled times: , , , , , . Find each relative frequency, correct to decimal places.
4. Using Q3’s data, which outcome had the largest positive difference from the theoretical ? Which had the largest negative difference?
5. Reasoning. Explain why relative frequency, calculated from many trials, is a better estimate of an unknown probability than the result of a single trial.
6. Reasoning. Describe two things that must stay the same across every trial of a chance experiment for the results to be trustworthy.
7. Challenge. In trials, an event occurred with relative frequency . The next trials all result in the event not occurring. Find the new overall relative frequency.

Answers: 1. Student’s own — the proportion of trials in which an event occurred, calculated as frequency ÷ total trials. 2. , compared with the theoretical — slightly higher. 3. : ; : ; : ; : ; : ; : . 4. Largest positive: face (); largest negative: face (). 5. A single trial can land on any outcome regardless of the true probability, so it tells you very little on its own; many trials average out this randomness, so the relative frequency settles closer to the true probability. 6. The physical method (e.g. toss height, spin force) and the equipment itself (e.g. the same coin, die or spinner) must stay identical across every trial. 7. Occurrences ; total trials ; new relative frequency .