At a glance. One lesson of about 60 minutes. Experimental probability as relative frequency, comparing with theory, the effect of the number of trials, and the “due” fallacy.
Calculator: no calculator. Every count is chosen so that the fraction is a terminating decimal and can be found by hand, so the work is in the meaning of the ratio.
What students will learn
Know
- Experimental probability is successes divided by trials.
- It is also called the relative frequency.
- Combining runs means adding counts, not averaging fractions.
Understand
- A short run can be far from the theory.
- More trials tend to settle near the theory, not exactly on it.
- Past results do not change the next one.
Do
- Find a relative frequency from a count or a table.
- Compare experimental and theoretical probability.
- Judge which of two runs is more reliable.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: finding relative frequencies and comparing with theory (questions 2 to 5).
- Criterion B. Investigating patterns: with the figure, describe how the relative frequency changes as rolls are added and say roughly after how many rolls it stops swinging widely.
- Criterion C. Communicating: stating a conclusion about fairness with reference to the number of trials.
- Criterion D. Applying mathematics in real-life contexts: lateness, drawing pins and games, and deciding when an experiment is long enough.
The one thing to do with the figure
Start at 5 rolls and ask what the line will do before each step to 20, 50, 100. Most students predict that it will head straight for the dashed line. It does not: it overshoots, comes back, and ends close to it.
Then ask: if we ran it again, would the line be the same? (Here, yes, because the run is fixed; a real die would give a different line, with the same settling shape.) The final gap, 0.15 against 0.167, is worth leaving unexplained as a prompt for the “what if” in Lesson 6.
Notice the four sixes in a row, at rolls 14 to 17. That is a streak in a fair run, not a sign of an unfair die.
A possible order
| 5 min | Warm-up, then answers. |
|---|---|
| 10 min | Hook: Zoe and Mo. Take a vote, then ask what evidence each would need. |
| 15 min | The figure; the key idea in their own words. |
| 10 min | Worked examples; the spinner in pairs. |
| 5 min | Question 1, the check. |
| 10 min | Your turn, Levels 1 to 3. |
| 5 min | Exit ticket, question 9. |
The answers
| Warm-up | 2/5; 0.4; 30. |
|---|---|
| Worked examples | Pin: 0.36. Coin: 0.46, difference 0.04. |
| Faded example | 0.35 and 35%. |
| 1. Relative frequency | B, 7/20. C gives the theoretical probability, not the experimental one. |
| 2. Decimal from a count | 0.4 = 10/25. |
| 3. Percentage from a count | 15 (15%). |
| 4. From a frequency table | 0.24 = 12/50. |
| 5. Experiment against theory | 0.05. |
| 6. No memory | A. C is the “due” slip from the hook; D reads a short run as a fact about the die. |
| 7. Planning more trials | 8 heads. |
| 8. Combining runs | 0.38 = 38/100. |
| 9. Which estimate to trust | B. Group B's run is 20 times longer. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| 10 or 2.5 in question 2 | Gave the count, or inverted the fraction. | Ask “out of how many?” and have them say the fraction before dividing. |
| 0.15 or 85 in question 3 | Left it as a decimal, or gave the complement. | Read the last words of the question together. |
| 0.6 in question 4 | Compared blue with green only. | Cover the colours: how many spins altogether? |
| 3 in question 5 | Subtracted counts instead of probabilities. | Ask: is the question about counts or about probabilities? |
| 1/2 or “more than 1/6” in question 6 | Treated six and not six as equally likely, or thought a six was due. | Roll a real die: does it know what came before? |
| 0.425 in question 8 | Averaged two relative frequencies from runs of different length. | Ask for the counts: how many events in each run? |
Questions worth asking
- Is the relative frequency the same as the probability? No: it estimates it. Listen for “gets closer with more trials”.
- Why can't a result be due? Listen for “the die doesn't remember”.
- How many trials are enough? There is no fixed number. Ask what difference they would call too large.
Differentiation
A way in for anyone stuck
- Give a pre-made tally chart so students only need to form the fraction.
- Have students run a real 20-roll experiment first and then compare with the figure.
- Provide a conversion strip for fractions out of 20, 25, 40 and 50.
Stretch
- A fair die is rolled 6 times. Without calculating, would you expect exactly one six? Roughly how often does it happen? (About 40% of the time.)
- Design an experiment to decide whether a drawing pin is equally likely to land point up or down. How many drops would you ask for, and why?
Students learning in English as an additional language
Pair each word with a motion: a thumb up for “happened”, an open hand for “out of how many”.
| Word | Plain meaning | Example |
|---|---|---|
| experimental probability | how often something happened out of the trials | 7 heads in 20 tosses is 7/20 |
| relative frequency | another name for experimental probability | 0.35 |
| trial | one go at the experiment | one roll of the die |
| theoretical probability | the probability worked out from equally likely outcomes | 1/6 for a six |
| fair | with no outcome favoured over any other | a fair die |
Sentence starters for explaining aloud or in writing:
- It happened … times out of … trials, so the relative frequency is …
- The experiment is close to / far from the theory because …
- I trust this result more because it has more …
Two things not to say
“It should even out.” That sounds like a promise that the next rolls will make up the difference. Say “the relative frequency tends to settle”.
“The experiment is wrong.” An experiment cannot be wrong; it can be short. Ask what a longer one might show.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- Relative frequencyA seed germinated in 17 of 20 trials. What is the experimental probability?
17/20 = 0.85. - PercentageA goal was scored from 8 of 40 shots. What is the experimental probability as a percentage?
20%.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ComparisonA die shows a six 30 times in 150 rolls. What is the difference from 1/6?
30/150 = 0.2; 1/6 ≈ 0.167, so about 0.033 above. - Earlier work (theory)A card is picked from 1 to 25. What is P(multiple of 5)?
5/25 = 1/5.
3Challenge
Reasoning, generalizing and working backwards.
- Planning9 heads in 20 tosses. How many heads are needed in the next 30 tosses for exactly 0.5 overall?
16. - ReliabilityA shop says its machine fills a cup correctly 90% of the time, from 10 tests. Why might you want more tests?
10 tests is a short run, so the estimate can be far from the truth.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The slider works with the keyboard and by touch; “Show me the steps” gives four cases in words.
On paper: a tally chart from a real class experiment does the same job as the figure.