Mathematics • Year 7 • Unit 4 • Lesson 16
Experimental Probability
Build fluency with the relative-frequency formula: experimental P(event) = frequency ÷ total trials. Use real experiment data to estimate probability and explain why bigger samples beat smaller ones every time.
1. I do — fully worked example
Read every line. Each step shows what to ask and why the calculation works.
Problem. A drawing pin is tossed 80 times and lands "point up" 18 times. Find the experimental probability of landing point up.
Step 1 — Read off the frequency and the total trials.
Frequency of "point up" = 18 Total trials = 80
Reason: the frequency is the count of the event we care about. The total trials is the count of every toss — not the number of possible outcomes.
Step 2 — Apply the formula.
Experimental P(point up) = frequency ÷ total = 18 ÷ 80
Reason: this is the definition of experimental probability (relative frequency).
Step 3 — Simplify and convert.
18/80 = 9/40 = 0.225 = 22.5%
Reason: divide top and bottom by 2. The decimal and percent are exact equivalents.
Answer: Experimental P(point up) = 9/40 = 0.225.
2. We do — fill in the missing steps
A die is rolled 50 times. The face "6" appears 9 times. Find experimental P(6) and compare to theory. Fill in each blank. 4 marks
Step 1 — Identify f and n.
f = _______ n = _______
Step 2 — Apply the formula.
Experimental P(6) = ___ ÷ ___ = _______
Step 3 — Compare to theoretical P(6).
Theoretical P(6) = 1 ÷ 6 ≈ _______ (3 dp)
Step 4 — Explain in one sentence.
The experimental value is close to theoretical because _______________________________________________
3. You do — independent practice
Show your formula line (f ÷ n), the simplified fraction, and the decimal to 3 dp where useful.
Foundation — apply the formula
3.1 In 80 coin flips, heads appeared 38 times. Find experimental P(heads). 1 mark
3.2 A spinner with red/blue/green sectors was spun 200 times. Red came up 84 times. Find experimental P(red). 1 mark
3.3 A bag of marbles is sampled 25 times (with replacement). Blue appears 7 times. Find experimental P(blue) as a fraction and decimal. 2 marks
3.4 A four-colour spinner is spun 60 times. Frequency table: Red 18, Blue 12, Green 15, Yellow 15. Check that the four relative frequencies add to 1. 2 marks
Standard — compare to theory
3.5 A fair coin is flipped 200 times and lands tails 92 times. (i) Find experimental P(tails). (ii) Compare with theoretical P(tails) = 0.5. (iii) Is the difference surprising? 3 marks
3.6 A die is rolled 600 times. Theoretically each face should appear 100 times. Face "3" appears 96 times. (i) Find experimental P(3). (ii) Is the difference of 4 normal variation or evidence of bias? Explain. 3 marks
Extension — push your thinking
3.7 Aria flips a coin 10 times and gets 7 heads. She says: "P(heads) = 0.7." Mira flips the same coin 1000 times and gets 503 heads. (i) Whose experimental value is more reliable, and why? (ii) Write one sentence using the words "law of large numbers". 3 marks
3.8 A frequency table for 200 rolls of a four-sided die shows {1: 48, 2: 52, 3: 46, 4: 54}. Calculate all four relative frequencies, check they sum to 1, then write one sentence about whether the die appears fair. 3 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — Die rolled 50 times (We do)
Step 1: f = 9, n = 50.
Step 2: Experimental P(6) = 9 ÷ 50 = 9/50 = 0.18.
Step 3: Theoretical P(6) = 1 ÷ 6 ≈ 0.167.
Step 4: The values are close because 50 trials is enough for the relative frequency to start settling near the theoretical value, but small variation is normal.
3.1 — 80 coin flips, 38 heads
P(heads) = 38 ÷ 80 = 19/40 = 0.475.
3.2 — Spinner, 200 spins, 84 red
P(red) = 84 ÷ 200 = 21/50 = 0.42.
3.3 — Bag of marbles, 25 draws, 7 blue
P(blue) = 7 ÷ 25 = 7/25 = 0.28.
3.4 — Four-colour spinner, 60 spins
Red 18/60 = 0.30; Blue 12/60 = 0.20; Green 15/60 = 0.25; Yellow 15/60 = 0.25. Sum = 0.30 + 0.20 + 0.25 + 0.25 = 1.00. ✓
3.5 — 200 flips, 92 tails
(i) P(tails) = 92 ÷ 200 = 23/50 = 0.46. (ii) Theory says 0.5; difference = 0.04. (iii) Not surprising — with only 200 trials, getting 92 instead of 100 tails is well within normal random variation.
3.6 — 600 rolls, 96 threes
(i) P(3) = 96 ÷ 600 = 4/25 = 0.16 (theory is ≈ 0.167). (ii) A difference of 4 out of 100 expected is normal variation — far too small to call the die biased. We would need a large difference (e.g. 50 vs 100) with this many trials to suspect bias.
3.7 — Whose value is more reliable?
(i) Mira's value (0.503) is more reliable because she ran far more trials. With only 10 flips Aria's result is easily distorted by random chance. (ii) "By the law of large numbers, as the number of trials increases the experimental probability gets closer and closer to the theoretical probability of 0.5."
3.8 — Four-sided die, 200 rolls
1: 48/200 = 0.24; 2: 52/200 = 0.26; 3: 46/200 = 0.23; 4: 54/200 = 0.27. Sum = 1.00. ✓ The four relative frequencies are all close to the theoretical 0.25, so the die appears fair within normal variation.