SkillrHub • Year 7 Maths

AC9M7P02 • Homework

Repeated Chance Experiments and Simulations

Problem-only student section: calculate, compare and interpret repeated-trial results. Show working. Answers are available in the checking section.

5 × 2 marks
Problems
5 × 5 marks
Multi-step problems
1 enrichment
Real vs simulation

Part A — 2-mark problems

  1. A coin-toss simulator runs 50 trials and shows 22 heads. Calculate the relative frequency of heads. (2)
  2. A fair die is rolled 120 times. Predict how many times a 6 should appear. (2)
  3. A spinner with 4 equal colours is spun 80 times. Predict how many times green should appear. (2)
  4. A digital die simulator runs 200 trials and shows 31 fives. Calculate the relative frequency of rolling a 5. (2)
  5. A fair coin is tossed 30 times and lands on tails 11 times. Compare the observed frequency with the expected frequency. (2)

Part B — 5-mark problems

  1. A fair-die simulator runs 300 trials: 1→44, 2→52, 3→49, 4→51, 5→54, 6→50. Calculate RF(5), compare it with 1/6, and state whether the result is reasonably consistent with the model. (5)
  2. A fair-coin simulator runs 500 trials: Heads 241, Tails 259. Calculate RF(heads), give the theoretical expected heads in 2000 trials, then use the observed RF to give an experimental prediction for 2000 trials. (5)
  3. A spinner with 5 equal sections is spun 400 times: R78, B82, G75, Y85, P80. Calculate RF(Yellow), compare with the theoretical probability and expected count, and explain the difference. (5)
  4. A two-dice simulation runs 600 trials. The sum is 8 on 78 trials. Calculate the relative frequency, compare with theoretical probability 5/36, and comment on the agreement. (5)
  5. A chance game is accurately sourced and played 200 times. A specified event occurs 56 times. Calculate its relative frequency and use it to estimate the event count in 800 similar trials. Explain why the estimate is not guaranteed. (5)

Part C — Enrichment

11. Real experiment vs digital simulation. A class records 139 heads in 300 real fair-coin tosses and 2488 heads in 5000 simulated fair-coin tosses. Calculate both relative frequencies, compare their distances from 0.5, predict the theoretical number of heads in 20,000 simulated tosses, and explain what this comparison does — and does not — show about large trial numbers and digital simulations.

Answers

1

22/50=0.44.

2

120×1/6=20.

3

80×1/4=20.

4

31/200=0.155.

5

Expected tails=15; observed=11, which is 4 fewer than expected.

6

RF(5)=54/300=0.18; theoretical≈0.1667; difference≈0.0133. This is a small difference consistent with random variation.

7

RF(H)=241/500=0.482. Theoretical prediction=1000 heads. Experimental-RF prediction=964 heads.

8

RF(Y)=85/400=0.2125; theoretical=0.2; expected count=80; observed is 5 above expected due to chance variation.

9

78/600=0.13; 5/36≈0.1389; difference≈0.0089, so the simulation is reasonably close to theory.

10

RF=56/200=0.28; estimate in 800=224. Random variation means 224 is not guaranteed.

11

Real RF=139/300≈0.4633; simulation RF=2488/5000=0.4976. Distances from 0.5 are about 0.0367 and 0.0024. Theoretical expected heads in 20,000=10,000. The larger simulation is closer in this example, consistent with long-run stabilisation, but this does not prove digital trials are inherently superior: trial count and model validity both matter, and randomness remains.