126i - Conducting Chance Experiments

Lesson 126 — Conducting Repeated Chance Experiments

Strand: Probability | Descriptor: AC9M7P02 | Duration: 45 minutes

Equipment: coins, dice, drawing pins or paper cups (one per pair), tally sheets.

Learning Intentions

• To conduct a repeated chance experiment and record results systematically.
• To observe how relative frequency behaves as the number of trials increases.

Success Criteria

I can:

1. Conduct a chance experiment with a consistent procedure.
2. Record results in a running tally with cumulative relative frequency.
3. Describe how relative frequency changes as trials increase.
4. Estimate a probability that cannot be calculated theoretically.

Warmup

(6 minutes — predict the run, mini whiteboards)

A fair coin will be tossed times.

1. Predict the number of heads.
2. Is heads guaranteed?
3. Which is more likely: exactly heads, or not exactly ?
4. Now predict for tosses. Would you be more or less confident of getting close to half?

Answers: 1. About ; 2. No; 3. Not exactly is more likely — getting precisely happens under a quarter of the time; 4. More confident of the proportion being close to , less confident of hitting the exact count.

Q3 surprises most classes. The expected value is the most likely single result, but it is still unlikely on any given run. Q4 is the lesson: the proportion stabilises even as the exact count becomes harder to hit.

Activities

Activity 1 — The Coin Experiment (16 min)

Pairs. The lesson’s core: watching relative frequency settle.

Procedure. Toss a coin times. After every tosses, pause and record the cumulative results.

After	Heads so far	Tosses so far	Relative frequency

The word “cumulative” is essential. Each row counts all heads so far, not just the last ten. Model one row on the board before starting.

Then, class pooling. Add every pair’s results to a class total, and compute the class cumulative relative frequency at , , and tosses.

Total tosses	Total heads	Relative frequency
(one pair)
(four pairs)
(ten pairs)
Whole class

The observation to draw out: individual pairs’ values scatter widely — , , . The pooled value converges toward and stays there. More trials, less scatter.

Circulating prompts:

Prompt	Purpose
Is your relative frequency getting closer to , or jumping about?	Individual runs jump; the trend is what matters.
Did your first predict your last ?	No — each toss is independent (pre-empts Lesson 129).
Are you counting cumulatively?	The commonest procedural error.
Whose result is furthest from ? Why is that not alarming?	Small samples vary; this is expected.

Activity 2 — The Unknown Probability (16 min)

Pairs. An experiment where theory offers no answer.

Choose one:

• A drawing pin. Drop it times from a fixed height. Record point-up or point-down.
• A paper cup. Drop it times. Record: lands upright / upside down / on its side.

Then:

1. Record a running tally in blocks of , with cumulative relative frequency.
2. Estimate each outcome’s probability from your final relative frequency.
3. Check your estimates total .
4. Could you have predicted these probabilities without experimenting? Explain.
5. Pool with another pair. Do the estimates change?

Socratic scaffolding:

Prompt	Purpose
Before dropping: predict the three cup probabilities.	Records an expectation to test.
Why can’t we count outcomes like a die?	The cup is not symmetric — the three results are not equally likely.
So how do we find the probabilities?	Only by experiment — experimental probability (Lesson 123).
Does your estimate improve with trials rather than ?	Yes — pooled data is more reliable.
Would another class get exactly your numbers?	No, but they should be close. The true probability is fixed; estimates vary around it.
What would change the probabilities themselves?	A different cup, a different drop height, a different surface — the procedure is part of the experiment.

The point to name: most real-world probabilities — weather, machine failure, medical outcomes — are of this kind. They cannot be counted from symmetry; they must be estimated from data.

Activity 3 — Inquiry: whose Result is Right? (7 min)

Whole class, using the pooled coin data.

Around the room, individual pairs recorded relative frequencies ranging from about to . The class total came out near .

1. Whose result was “correct”?
2. Why did the class total behave differently from the individuals?
3. If one pair got , does that mean their coin was biased?
4. How many tosses would you want before claiming a coin was unfair?

Discussion targets: every pair’s result was correctly recorded — none was wrong. Individual runs vary because trials is a small sample; pooling to averages the variation away. A single result of from tosses is entirely ordinary. Claiming bias would want hundreds or thousands of tosses with a persistent deviation.

The law being observed, named: as the number of trials grows, relative frequency approaches the theoretical probability. This is why insurers, casinos and quality controllers can predict aggregate outcomes precisely while individual events stay unpredictable (Lesson 123 E5).

Checks for Understanding

(5 minutes — exit ticket, collected with the tally sheet)

1. State your final relative frequency of heads over tosses.
2. State the class’s pooled relative frequency.
3. Which is closer to , and why would you expect that?
4. Why can a drawing pin’s probability not be calculated theoretically?
5. Reasoning. A pair got heads in tosses. Should they conclude their coin is biased? Explain.

Answers: 1–2. Student’s own; 3. The pooled figure, because more trials reduce the influence of chance; 4. It is not symmetric, so there is no reason to treat point-up and point-down as equally likely — only experiment can estimate it; 5. No. of is a relative frequency of , above the expected , but well within the variation seen across the room. Many more tosses would be needed.

Common Misconceptions

Misconception	How to pre-empt it
Recording block totals instead of cumulative ones.	Modelled on the board; checked while circulating.
Believing a deviating result means faulty equipment.	The inquiry — every pair’s result was correct.
Expecting relative frequency to reach exactly .	It approaches; it need never land exactly.
Thinking earlier tosses influence later ones.	Flagged here, treated fully in Lesson 129.
Assuming every probability can be calculated from theory.	The pin and cup experiments.
Ignoring the procedure’s role in the probability.	Drop height and surface change the pin’s probability.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A coin tossed times gives heads. Find the relative frequency and its difference from the theoretical probability.

Answer

; a difference of from — small and unremarkable.

E2 (AMC Junior style). Four pairs report relative frequencies of , , and , each from tosses. Find the pooled relative frequency.

Answer

Heads: from tosses, so exactly. Pooling averages out the individual scatter.

E3 (Challenge). A drawing pin lands point-up times in drops, then times in the next . What is the best single estimate of ?

Answer

Pool everything: . Using all the data gives a better estimate than either run alone.

E4 (Challenge). Two classes drop identical pins from different heights and get and . Are they contradicting each other?

Answer

Not necessarily — drop height is part of the experimental procedure, so the two classes may be measuring genuinely different probabilities. Comparing results requires comparing procedures first.

E5 (Challenge). Why does pooling tosses from twenty pairs give a better estimate than one pair tossing times twenty times over?

Answer

It doesn’t — they are equivalent, provided the procedure is identical. Both give trials, and it is the number of trials that matters, not who performed them. (A useful check on whether students understand what is doing the work.)

Homework

1. Explain in one sentence what “relative frequency” means.

2. A coin tossed times gives heads. (a) Relative frequency. (b) Comparison with theory. (c) Is this surprising?

3. From this cumulative record, complete the relative frequency column:

   Tosses	Heads	Relative frequency

4. Describe in one sentence what happens to the relative frequency in Q3 as the tosses increase.

5. A pin dropped times lands point-up times. (a) Estimate . (b) Estimate . (c) Check they total .

6. Conduct your own experiment at home: drop a small object (a bottle cap, a coin, a pen lid) times and record two possible outcomes. Present a cumulative tally table and estimate each probability.

7. Explain why your Q6 probabilities are estimates rather than exact values.

8. Reasoning. Explain why a class’s pooled result is more reliable than one pair’s.

9. Reasoning. A student says “I got heads in , so the coin must be biased.” Explain what is wrong with this reasoning.

10. Challenge. Two students each toss a coin times, getting and . A third claims “the coins cancel out, so both are fair.” Assess this reasoning.

Answers: Q2 — (a) (b) above the theoretical (c) not surprising — against an expected is ordinary variation in tosses. Q3 — , , , , . Q4 — it starts high and settles progressively closer to as trials increase. Q5 — (a) (b) (c) ✓. Q7 — the object is not symmetric, so no theoretical calculation is available; each estimate is based on a limited number of trials and would shift slightly with more. Q8 — more trials reduce the influence of chance on the proportion, so the pooled figure lies closer to the true probability. Q9 — of is , above expectation, but a fair coin produces results like this routinely; one short run cannot establish bias. Q10 — the conclusion (both fair) is plausible, but the reasoning is wrong: results do not “cancel out”, and each coin must be judged on its own data. Pooling them would only be valid if they were the same coin.