Mathematics • Year 10 • Unit 4 • Lesson 18
Theoretical vs Experimental Probability — Skill Drill
Build fluency with Lesson 18: theoretical probability (from assumptions), experimental probability (= number of successes / total trials), and the law of large numbers (experimental → theoretical as trials increase). Also drill the independent vs dependent test.
1. I do — fully worked example
Each step shows the rule on the right.
Problem. A fair six-sided die is rolled 60 times. A 6 comes up 8 times. (a) Calculate the experimental P(6). (b) State the theoretical P(6). (c) Use the law of large numbers to explain why 8 is "slightly less than expected".
Step 1 — Experimental probability formula.
Exp P(6) = successes / total trials = 8 / 60 = 2/15 ≈ 0.133.
Reason: Lesson 18 Key Term — "Relative frequency = number of successes / total trials".
Step 2 — Theoretical probability of a 6.
Theor P(6) = 1/6 ≈ 0.167.
Reason: 6 equally likely faces; 1 favourable face.
Step 3 — Expected number of 6s in 60 rolls.
Expected = 60 × 1/6 = 10.
Reason: theoretical probability × number of trials.
Step 4 — Apply the law of large numbers.
Observed (8) < expected (10), but only by 2. With more rolls, the proportion 8/60 ≈ 0.133 should drift back toward 1/6 ≈ 0.167.
Reason: Lesson 18 Key Term — "Law of large numbers: as trials increase, experimental probability approaches theoretical probability."
Answer: Exp P(6) = 2/15; theoretical P(6) = 1/6; observed result is slightly under expected but consistent with random variation in a small number of trials.
2. We do — fill in the missing steps
Same structure as Section 1, with the working faded. 4 marks
Problem. A fair coin is flipped 10 times and lands on heads 7 times. (a) Find the experimental P(heads). (b) State the theoretical P(heads). (c) Predict what will happen to the experimental P(heads) if the coin is flipped 1000 times instead.
Step 1 — Experimental P(heads).
Exp P(heads) = ______/10 = ______.
Step 2 — Theoretical P(heads).
Theor P(heads) = ____ (fraction).
Step 3 — Compare. Difference = ______ − 0.5 = ______. After only 10 flips the difference is __________ (small / large).
Step 4 — Predict with more trials. As the number of trials grows from 10 to 1000, the law of large numbers says the experimental P(heads) will __________________________________.
3. You do — independent practice
Show working. Foundation = single-step calc. Standard = compare expected vs observed. Extension = independence/dependence trap.
Foundation — apply the formulas
3.1 A spinner is spun 80 times and lands on "win" 32 times. Find the experimental P(win). 1 mark
3.2 A fair four-sided die is rolled. The theoretical P(rolling a 1) = ______. 1 mark
3.3 200 lightbulbs are tested; 6 are faulty. Find the experimental P(faulty). 1 mark
3.4 A fair coin is flipped 250 times. The expected number of heads is ______. 1 mark
Standard — compare expected and observed
3.5 A bag of 20 jellybeans contains 5 red, 7 yellow, 8 green. The bag is "tested" by selecting one bean 60 times with replacement. The red bean is selected 18 times. (a) Find the experimental P(red). (b) State the theoretical P(red). (c) Are the two close? In one sentence, apply the law of large numbers. 3 marks
3.6 Use the multiplication rule for independence (Lesson 18 Remember card): a coin is flipped and a fair die is rolled. Show that the events are independent, then compute P(tail and a 6). 2 marks
Extension — the independence test
3.7 A bag contains 5 red and 5 blue marbles. Two marbles are drawn without replacement. Find P(first red) and P(second red | first red). Use Lesson 18's "P(A|B) = P(A)?" test to decide whether the events are independent. 3 marks
3.8 A friend says: "Independent events can never both happen at the same time." Apply Lesson 18's misconception fix to write a one-sentence correction, and give one example of independent events that DO both occur. 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (coin: 7 heads in 10 flips)
Step 1: Exp P(heads) = 7/10 = 0.7.
Step 2: Theor P(heads) = 1/2 (= 0.5).
Step 3: difference = 0.7 − 0.5 = 0.2. After only 10 flips, this is large (small samples have a lot of variation).
Step 4: The experimental P(heads) will approach 0.5 (the theoretical value) — the law of large numbers.
3.1 — Spinner
Exp P(win) = 32/80 = 2/5 = 0.4.
3.2 — Theoretical for a tetrahedron
P(1) = 1/4.
3.3 — Faulty lightbulbs
Exp P(faulty) = 6/200 = 3/100 = 0.03.
3.4 — Expected heads
250 × 1/2 = 125.
3.5 — Jellybean test
(a) Exp P(red) = 18/60 = 3/10 = 0.3.
(b) Theor P(red) = 5/20 = 1/4 = 0.25.
(c) The two are close (0.3 vs 0.25). As trials grow, the experimental value will move closer to 0.25 — law of large numbers.
3.6 — Independent coin and die
The coin and die do not affect each other (mechanically separate), so P(tail | a 6) = P(tail) = 1/2 — the events are independent. P(tail and 6) = 1/2 × 1/6 = 1/12.
3.7 — Without-replacement test
P(first red) = 5/10 = 1/2. After a red is removed, 4 red remain in 9. P(second red | first red) = 4/9 ≈ 0.444.
Since P(second red | first red) ≠ P(second red) = 1/2, the events are not independent (dependent). The bag has changed after the first draw.
3.8 — Independence misconception
Wrong. Independent events CAN both occur — independence simply means the outcome of one does not change the probability of the other. Example: flipping a head AND rolling a 6 are independent and both happen together with probability 1/12.