At a glance. One lesson of about 60 minutes. It assumes theoretical probability and the tree and table work in this unit. Students find relative frequency, expected frequency and combined estimates, and explain why a short run proves little.
Calculator: GDC optional. A GDC or spreadsheet can simulate thousands of trials quickly, which is the best way to see settling. The optional section shows the random-integer command; the questions themselves are short calculations.
What students will learn
Know
- Relative frequency = times it happened ÷ trials.
- Expected frequency = P × n.
- Larger samples give more reliable estimates.
Understand
- Randomness has long runs and uneven short samples.
- Independent trials have no memory, so nothing is due.
- An experimental probability is an estimate of the theoretical probability and varies from sample to sample.
Do
- Find a relative frequency and combine two samples.
- Predict a frequency from a probability, or estimate a probability from a frequency.
- Say what a given result does and does not show.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: relative frequency and expected frequency (questions 1 to 8).
- Criterion B. Investigating patterns: both figures. Ask students to predict the longest run in a sequence before it is shown, and how far apart the runs will be at 1000 flips.
- Criterion C. Communicating: justifying a conclusion from a sample, with the sample size stated (the check and question 9).
- Criterion D. Applying mathematics in real-life contexts: quality testing, forecasts and what a result in a small sample does or does not show.
The one thing to do with the figure
Make students commit before they slide. In the first figure, hide the run and ask “What is the longest run of the same face in this sequence?” Most students say 2 or 3. Then move the slider and reveal it.
In the second figure, ask “How far apart will the two runs be at 1000 flips?” and collect guesses before sliding. Then ask whether more flips would put them exactly together.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 7 min | Hook: the teacher's trick. Collect predictions about runs in real and invented lists. |
| 10 min | First figure and key idea. Settle the hook. |
| 10 min | Second figure, the warning card and the worked example; pairs finish the faded example. |
| 5 min | The check. If many choose A, return to the sequences in the first figure. |
| 18 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 9. |
The answers
| Warm-up | 1/2; 70; 3/4. |
|---|---|
| Hook | Real lists usually contain a run of 5 or more (about 4 times in 5; sequence 1 in the figure is one of the exceptions); invented lists rarely have runs longer than 3 or 4, because people think long runs look unfair. |
| Worked example | 0.62 and 248 point-up drops. |
| Faded example | 0.6, and 30 shots. |
| 1. Small samples vary | B. A trusts a tiny sample; C expects an exact split; D is the gambler's fallacy. |
| 2. Relative frequency | 0.25. 60 ÷ 240. |
| 3. Expected frequency | 24. 0.4 × 60. |
| 4. Using an estimate to predict | 192. 0.024 × 8000. |
| 5. Combining two experiments | 0.46. 92 ÷ 200. |
| 6. Independence of flips | D. A thinks the run continues; B is the gambler's fallacy; C ignores that the coin is fair. |
| 7. Simulating with random digits | 3. 3, 0 and 6. |
| 8. Working back from an expected frequency | 150. 12 ÷ 0.08. |
| 9. What a result does and does not say | C. A treats the sample as the true probability; B treats the sample as a promise; D over-reads a small sample. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Trusts a tiny sample, expects an exact split, or thinks the coin evens out. | Ask: how often does a fair coin give 5 or 6 heads in 8? Run it 10 times. |
| 4 or 0.75 in question 2 | Inverted the fraction, or gave the complement. | Name the top number as “times it happened” and the bottom as “trials”. |
| 36 or 150 in question 3 | Gave the dry days, or divided. | Check the answer against the probability: about 40% of 60 is less than 30. |
| 16 or 96 000 in question 4 | Used the sample size ratio, or multiplied the count by the batch size. | Find the proportion faulty first as a decimal. |
| 0.64, 0.4 or 0.52 in question 5 | Used one batch or averaged two proportions. | Total the heads and the flips before dividing. |
| A, B or C in question 6 | Thinks a run continues, thinks heads is due, or ignores the fair coin. | Run 20 flips and ask what follows each run of tails. |
| 2 or 4 in question 7 | Used the wrong number of digits for 70%. | List the digits that mean made: 0, 1, 2, 3, 4, 5, 6. How many is that? |
| 0.96, 96 or 1.5 in question 8 | Multiplied, or took 0.08 as 8. | Check: 150 plays at 0.08 is 12 prizes. |
| A, B or D in question 9 | Treats a sample as the truth, as a promise, or as proof of bias. | Ask: if the die were fair, what would you expect in 40 rolls? Is 11 far from that? |
Questions worth asking
- Why does the proportion settle even though the count of heads need not? Listen for “the total gets bigger, so a gap of a few flips matters less as a proportion”.
- Is 5 tails in a row evidence that a coin is biased? Not by itself: runs of 5 are expected in 50 flips.
- Which gives a better estimate of a probability: 20 trials or 500? 500: the relative frequency varies less from sample to sample.
- What is the difference between experimental and theoretical probability? Theoretical comes from equally likely outcomes; experimental comes from what happened.
Differentiation
A way in for anyone stuck
- Give a table to fill in: number of flips, number of heads, relative frequency, for several batch sizes.
- Use a physical coin or die in pairs before the figures, recording each batch on a tally chart.
- Write the formula in words on the board: times it happened ÷ trials.
Stretch
- Design a simulation with random digits for a spinner with sectors of probability 0.2, 0.3 and 0.5, and run 20 trials.
- A coin is flipped 100 times. Roughly how many heads would make you doubt the coin is fair? Defend your threshold.
Students learning in English as an additional language
“Trial” is one go. “Relative frequency” means how often something happened compared with how many tries. “Expected” means what you would predict on average, not what must happen. “Due” is a word for something that is owed, which a coin never owes.
| Word | Plain meaning | Example |
|---|---|---|
| trial | one repeat of an experiment | one spin of a spinner |
| relative frequency | the number of times an event happens divided by the number of trials | 7 heads in 20 flips is 0.35 |
| experimental probability | a probability estimated from results | a cup landing upright in 13 of 50 throws |
| theoretical probability | a probability worked out from equally likely outcomes | 1/2 for a fair coin |
| expected frequency | the probability multiplied by the number of trials | 0.2 × 50 trials |
| random | unpredictable one trial at a time, but with a pattern in the long run | the face of a fair die |
Sentence starters for explaining aloud or in writing:
- The relative frequency is … out of …, which is ….
- I would expect about … because ….
- This sample is / is not large enough to say, because ….
Two things not to say
“It is due.” The coin has no memory.
“The probability says it will happen 6 times.” It says 6 is the average; any one run varies.
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.
- EstimateA drawing pin lands point up 26 times in 40 drops. What is the relative frequency?
26 ÷ 40 = 0.65. - ExpectP(a six) = 1/6. How many sixes do you expect in 120 rolls?
20.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA farm in Kenya finds 18 of 400 eggs cracked. How many cracked eggs would you expect in 1600 eggs?
18 ÷ 400 = 0.045, and 0.045 × 1600 = 72. - Earlier work (Lesson 5)P(A) = 0.5, P(B) = 0.3 and A, B are mutually exclusive. P(A or B)?
0.8.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningSam says: “I got 7 heads in 10 flips, so P(heads) = 0.7.” Give two reasons why this is weak.
Only 10 flips; another 10 would differ; a fair coin often gives 7 heads in 10. - Working backwardsA coin is flipped 80 times and the relative frequency of heads is 0.45. How many heads?
36.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard (tab to the slider, then the arrow keys) and by touch. Under each one, “Show me the steps” gives the same states in words.
No device? Give each pair a coin and have them record 20, then 40, flips and pool the class results on the board.