Mathematics • Year 7 • Unit 4 • Lesson 16
Experimental Probability — Mixed Challenge
Bring together relative frequency, the law of large numbers, expected counts, and bias judgement. Spot a classification mistake and design your own probability experiment.
1. Mixed problems
Show working. 2 marks each
1.1 A spinner is spun 240 times. The "win" sector comes up 78 times. Find experimental P(win) as a fraction in lowest terms and as a decimal to 3 dp.
1.2 A coin is flipped 1000 times and lands heads 487 times. (i) Find experimental P(heads). (ii) State whether the difference from 0.5 is evidence of bias and justify in one sentence.
1.3 A die is rolled 50 times and the frequencies are {1: 9, 2: 8, 3: 7, 4: 9, 5: 8, 6: 9}. Find experimental P(rolling an even number).
1.4 If P(make) for a free throw is estimated at 0.65 from 200 attempts, how many makes are expected in a season of 800 attempts?
1.5 Sarah and Tom both estimate the probability that the school bus is late. Sarah tracks 20 days and gets P(late) = 0.30. Tom tracks 100 days and gets P(late) = 0.18. Whose estimate is more trustworthy and why?
1.6 A drawing pin is tossed 400 times and lands "point up" 144 times. (i) Find experimental P(point up). (ii) Find experimental P(point down). (iii) Verify the two probabilities sum to 1.
2. Find the mistake
A Year 7 student has submitted the following working for the question: "A four-colour spinner is spun 80 times. Red appears 22 times. Find experimental P(red)." Exactly one line contains a serious error. Spot it, explain why it's wrong, then redo the calculation correctly. 3 marks
Student's working:
Line 1: The spinner has 4 colours, so total outcomes = 4.
Line 2: Red appeared 22 times, so frequency = 22.
Line 3: Experimental P(red) = frequency ÷ total outcomes
Line 4: = 22 ÷ 4 = 5.5. So P(red) = 5.5.
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Show the correct calculation for experimental P(red).
Stuck? Recall: probability is always between 0 and 1 — P = 5.5 is impossible. So something is wrong with the denominator.3. Open-ended challenge — design your own probability experiment
This question has many correct answers. Show your work clearly. 4 marks
3.1 Design a class-sized experiment (≈ 100 trials minimum) to estimate the experimental probability of a chosen event. You must give:
- the event you are investigating (e.g. "a bottle cap landing on its base"),
- a method: what one trial looks like, what counts as the event, and how many trials you'll run,
- a blank frequency table with columns "Outcome", "Frequency", "Relative frequency",
- a sentence on how you would make the estimate more reliable.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Spinner 240 spins, 78 wins
P(win) = 78 ÷ 240 = 13/40 = 0.325.
1.2 — Coin, 1000 flips, 487 heads
(i) P(heads) = 487 ÷ 1000 = 0.487. (ii) Not evidence of bias — 487 vs an expected 500 is well within normal variation for 1000 trials, and the value is very close to 0.5.
1.3 — Die, 50 rolls, even numbers
Frequency of evens = freq(2) + freq(4) + freq(6) = 8 + 9 + 9 = 26. P(even) = 26 ÷ 50 = 13/25 = 0.52.
1.4 — Free-throw season
Expected makes = 0.65 × 800 = 520 makes.
1.5 — Sarah vs Tom
Tom's estimate (0.18 from 100 days) is more trustworthy because more trials give a more reliable estimate — the law of large numbers. Sarah's 20-day sample is too small; the high "0.30" could easily reflect short-term traffic disruption rather than the true probability.
1.6 — Drawing pin, 400 tosses, 144 point up
(i) P(point up) = 144 ÷ 400 = 9/25 = 0.36.
(ii) P(point down) = 256 ÷ 400 = 16/25 = 0.64.
(iii) 0.36 + 0.64 = 1.00. ✓ The two outcomes cover every trial.
2 — Find the mistake
(a) The mistake is on Line 1 (and it carries through to the wrong denominator on Line 4).
(b) The denominator in experimental probability is the total number of trials (here, 80 spins), not the number of possible outcomes (4 colours). Using 4 gives an impossible answer above 1.
(c) Correct: P(red) = 22 ÷ 80 = 11/40 = 0.275.
3 — Design your own experiment (sample answer)
Event: a bottle cap landing "open side up" when tossed.
Method: One trial = drop the cap from 30 cm. Record whether it lands open side up, closed side up, or on its edge. Run 100 trials.
Frequency table (blank):
Outcome Frequency Relative frequency
Open up ____ ____ / 100
Closed up ____ ____ / 100
On edge ____ ____ / 100
Total 100 1.00
Reliability: Increase the number of trials (e.g. by combining results from every group in the class to get 2500 trials), or repeat in the same conditions on different days. By the law of large numbers, relative frequencies will settle closer to the true probability.
Marking: 1 mark for clear event, 1 for a workable method, 1 for a usable frequency table, 1 for a reliability improvement.