Mathematics • Year 8 • Unit 4 • Lesson 19
Experimental Probability
Build fluency with calculating experimental probability (frequency ÷ trials), comparing it to theoretical, and applying the law of large numbers and expected frequencies.
1. I do, fully worked example
Watch how we calculate experimental probability, compare to theoretical, and find an expected frequency.
Problem. A die is rolled 60 times. The outcome 4 appears 13 times. (a) Find the experimental P(rolling a 4). (b) State the theoretical probability. (c) How many 4s would we expect in 600 rolls of a fair die?
Step 1, Apply the experimental probability formula.
P(4) ≈ frequency / trials = 13 / 60
Reason: experimental probability is the fraction of trials where the event occurred.
Step 2, Convert to decimal for easier comparison.
13 / 60 ≈ 0.217 (3 d.p.)
Step 3, Find theoretical probability.
A fair die: P(4) = 1/6 ≈ 0.167
Reason: each face is equally likely, so 1 out of 6.
Step 4, Compare and comment.
0.217 vs 0.167 → close, but 60 trials is a small sample. The difference (0.05) is within normal variation.
Step 5, Expected frequency in 600 rolls.
Expected = P × n = (1/6) × 600 = 100
Answer: Experimental P ≈ 13/60 ≈ 0.217; theoretical P = 1/6 ≈ 0.167; expected 4s in 600 rolls = 100.
2. We do, fill in the missing steps
Same shape as Section 1, but with the working faded. Fill in each blank. 4 marks
Problem. A coin is flipped 80 times. Heads appears 36 times. (a) Find experimental P(H). (b) Compare to theoretical. (c) Expected heads in 800 flips of a fair coin.
Step 1, Experimental formula:
P(H) ≈ ______ / ______
Step 2, Decimal:
______ / ______ = ______ (3 d.p.)
Step 3, Theoretical P(H):
P(H) = ______ / ______ = ______
Step 4, Compare:
______ vs ______ → close / not close ? (circle)
Step 5, Expected H in 800 flips:
Expected = ______ × 800 = ______
3. You do, independent practice
Show working under each problem. Foundation tests the formula, standard does the comparison, extension uses expected frequencies and the law of large numbers.
Foundation, recall and formula
3.1 A spinner is spun 40 times and lands on red 18 times. Calculate the experimental P(red). Simplify. 1 mark
3.2 Define theoretical probability in one sentence. Use the words "equally likely". 1 mark
3.3 State the law of large numbers in your own words. 1 mark
3.4 A die is rolled 600 times. State the expected frequency of rolling a 2. Show the calculation. 1 mark
Standard, calculate and compare
3.5 A die is rolled 120 times. The outcome 5 appears 18 times. Calculate the experimental P(rolling a 5) and compare to the theoretical probability. Is there strong evidence of bias? 2 marks
3.6 A 4-sided die (faces 1, 2, 3, 4) is rolled 200 times with this frequency table: 1→55, 2→48, 3→52, 4→45. Calculate the experimental probability for each face. Does the die appear fair? 2 marks
Extension, expected frequency and reasoning
3.7 A bag has 5 marbles (2 Red, 3 Blue). One marble is drawn 300 times with replacement. (a) State the theoretical P(Red). (b) State the expected frequency of Red. (c) If 130 Reds were observed, is this consistent with a fair bag? Explain. 2 marks
3.8 Two students each roll a die 30 times. Student A gets 7 sixes. Student B gets 3 sixes. (a) Calculate each student's experimental P(6). (b) If they combine their data (60 trials total), calculate the combined P(6). (c) Which estimate is most reliable and why? 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2, We do (coin, 36 H in 80)
Step 1: P(H) ≈ 36 / 80. Step 2: 36/80 = 0.450. Step 3: Theoretical P(H) = 1/2 = 0.500. Step 4: close (within 0.05; 80 trials is a small sample so this variation is normal). Step 5: Expected = 0.5 × 800 = 400.
3.1, Spinner, 18 reds in 40
P(red) ≈ 18 / 40 = 9/20 = 0.45.
3.2, Theoretical probability
Theoretical probability is calculated by counting equally likely outcomes, favourable ÷ total, without running an experiment.
3.3, Law of large numbers
As the number of trials increases, the experimental probability gets closer to the theoretical probability.
3.4, Expected 2s in 600 rolls
Expected = (1/6) × 600 = 100.
3.5, Die rolled 120 times, 18 fives
Experimental P(5) = 18/120 = 3/20 = 0.150. Theoretical P(5) = 1/6 ≈ 0.167. The difference is small (about 0.017), well within normal sampling variation for 120 trials. No strong evidence of bias.
3.6-4-sided die fairness
P(1) ≈ 55/200 = 0.275; P(2) ≈ 48/200 = 0.240; P(3) ≈ 52/200 = 0.260; P(4) ≈ 45/200 = 0.225. Theoretical = 0.250 for each. All four are close to 0.250 (within about 0.025). The die appears fair the small variations are consistent with chance in 200 trials.
3.7, Bag of marbles, 300 draws
(a) P(Red) = 2/5 = 0.4.
(b) Expected Red = 0.4 × 300 = 120.
(c) 130 is close to the expected 120 (10 above). This is well within normal variation for 300 trials, consistent with a fair bag.
3.8, Two students, then combined
(a) A: P(6) = 7/30 ≈ 0.233. B: P(6) = 3/30 = 0.100.
(b) Combined: P(6) = (7+3)/60 = 10/60 = 1/6 ≈ 0.167.
(c) The combined estimate (60 trials) is more reliable, by the law of large numbers, more trials gives a value closer to the theoretical 1/6. Either individual estimate of 30 trials is noisy.