Mathematics • Year 10 • Unit 4 • Lesson 18
Theoretical vs Experimental — Mixed Challenge
Pull together Lesson 18: theoretical vs experimental probability, the law of large numbers, the independence test (P(A|B) = P(A)), and with-replacement vs without-replacement situations. Then spot a Year 10 mistake and design your own probability experiment.
1. Mixed problems
Decide which Lesson 18 idea each question is about before you start. 3 marks each
1.1 A spinner lands on red 24 times out of 80 spins. Find the experimental P(red) and predict how many reds you would expect in 200 more spins.
1.2 A coin is flipped and a die is rolled. State whether the two events are independent. Find P(head AND a number > 4).
1.3 A bag contains 4 red and 6 blue marbles. Two marbles are drawn with replacement. Show that the draws are independent, then find P(both red).
1.4 Repeat 1.3 but now without replacement. Use Lesson 18's misconception fix to explain in one sentence why the draws are now dependent. Find P(both red).
1.5 Explain in your own words what the law of large numbers means, using one example that is NOT a coin or a die.
1.6 A weather website claims a 30% chance of rain on a given day in Sydney during August. Over 31 days you record 12 rainy days. (a) State the theoretical P(rain) per day. (b) Calculate the experimental P(rain) for August. (c) Are the two close enough to support the 30% claim? Use the expected number 31 × 0.30 in your reasoning.
2. Find the mistake
A Year 10 student tries this: "A coin is flipped 8 times and lands on heads 3 times. What is the experimental P(heads), and what does the law of large numbers say will happen with more flips?" Exactly one line is wrong. 3 marks
Student's working:
Line 1: Experimental P(heads) = 3 / 8 = 0.375.
Line 2: Theoretical P(heads) for a fair coin = 0.5.
Line 3: The law of large numbers says that after many more flips, the experimental probability will EXACTLY equal 0.5.
Line 4: So with 1000 more flips we will see exactly 500 heads.
(a) Which line(s) contain the mistake?
(b) Explain why, naming the misconception from Lesson 18.
(c) Write a corrected statement.
Stuck? Lesson 20 misconception card: the law of large numbers says "approaches", not "exactly equals".3. Open-ended challenge — design and predict
Many valid answers. 4 marks
3.1 Design a probability experiment you could actually run in class to test a theoretical probability (e.g. drawing-pin landing point-up vs flat, M&M colours from a bag, randomly chosen pages of a textbook starting with a vowel, etc.). For your experiment, describe:
- (i) the variable being measured and the event of interest,
- (ii) the theoretical P (if you can predict one) OR a written prediction with justification,
- (iii) the number of trials you would run and WHY (link to the law of large numbers),
- (iv) how you would record the data (sketch a tally table),
- (v) two questions you would answer at the end (one numerical, one conceptual).
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Spinner
Exp P(red) = 24/80 = 3/10 = 0.3. Expected reds in 200 spins = 200 × 0.3 = 60.
1.2 — Coin and die
Independent (coin does not affect die). P(head) = 1/2; P(> 4) = 2/6 = 1/3. P(head and > 4) = 1/2 × 1/3 = 1/6.
1.3 — With replacement
With replacement → bag is unchanged → independent. P(red) = 4/10 = 2/5 each draw. P(both red) = 2/5 × 2/5 = 4/25.
1.4 — Without replacement
Without replacement, the bag changes after the first draw. Lesson 18 misconception fix: "Drawing without replacement creates dependent events because the probability changes after each draw."
P(both red) = 4/10 × 3/9 = 12/90 = 2/15.
1.5 — Law of large numbers (own example)
Any valid example. Sample: an AFL goalkicker's career conversion rate. In one game he might convert 1 of 5 attempts (20%), in another 4 of 5 (80%). But across 5,000 career kicks his average will settle close to his "true" conversion rate (say 65%). The law of large numbers says the experimental probability gets closer to the underlying rate as trials grow.
1.6 — Rain in Sydney
(a) Theor P(rain per day) = 0.30. (b) Exp P(rain in August) = 12/31 ≈ 0.387. (c) Expected days of rain = 31 × 0.30 = 9.3. Observed = 12, which is 2.7 above expected — a small difference within normal weather variation. One month of data is far too short for the law of large numbers to give a tight result; the 30% claim is not contradicted.
2 — Find the mistake
(a) The mistakes are on Lines 3 and 4 (Line 3 is the root cause; Line 4 follows from it).
(b) The law of large numbers says the experimental probability approaches the theoretical probability — not that it becomes exactly equal. Lesson 20 misconception card explicitly warns: "It does not guarantee exact equality." Getting exactly 500/1000 heads is unlikely; getting close to 500 is.
(c) Corrected: After many more flips, the experimental P(heads) will get closer and closer to 0.5, but the actual count of heads in 1000 more flips will probably be near 500 (say between 470 and 530) — not exactly 500.
3 — Open-ended challenge (sample solution)
Experiment: Drop a standard drawing pin onto a hard floor and record whether it lands point-up (PU) or on its side (S).
(i) Variable: landing orientation. Event of interest: "point up".
(ii) No clean theoretical probability — pin shape is asymmetric. Prediction: about 0.3 (just a guess based on shape).
(iii) Run 200 trials. The law of large numbers tells me that with only 20 trials I could be 0.20 off; with 200 my experimental P should be within about 0.05 of the true value, giving a reasonable estimate.
(iv) Tally table with rows "PU" and "S" and a Total row. Fill in tallies in groups of 5.
(v) Numerical: "What is the experimental P(point up)?" Conceptual: "If I doubled the number of trials, would my estimate definitely improve? Explain using the law of large numbers."
Marking: 1 mark — clear variable and event; 1 — sensible prediction with reason; 1 — number of trials justified via law of large numbers; 1 — tally table sketched + two valid questions.