Probability CardsANSWER KEY
Simulation lab: experiment vs theory
Ages 14+ · about 35 minutes
Goal: See how experimental frequency approaches theoretical probability and how error shrinks as samples grow.
Remember: Exact probability of one pair is 1,098,240 ÷ 2,598,960 = 42.26%. Error = |observed − exact| in percentage points.
Lab steps
- Open the five-card pattern simulator on the website (Simulators, then Five-card pattern frequencies).
- Type the seed classroom into the seed box.
- Choose 100 samples and press Run simulation. Write the count for One pair in the table below.
- Repeat for 1,000, 10,000 and 100,000 samples with the same seed.
- Complete the table: observed % = count ÷ samples × 100; error = |observed % − 42.26|.
| Samples | One pair count | Observed % | Exact % | Error (points) |
|---|---|---|---|---|
| 100 | 43 | 43.00 | 42.26 | 0.74 |
| 1,000 | 418 | 41.80 | 42.26 | 0.46 |
| 10,000 | 4,150 | 41.50 | 42.26 | 0.76 |
| 100,000 | 42,420 | 42.42 | 42.26 | 0.16 |
1.Complete the results table (it is printed above these questions).
Answer: See the answer key table: results with the seed 'classroom' are reproducible, so your numbers should match exactly.
2.Which sample size gave the smallest error? Is that what you expected?
Answer: Usually 10,000 or 100,000: more samples generally means less error, but a single run can be lucky or unlucky, so a smaller sample can occasionally beat a larger one.
3.The predicted standard error is √(p(1 − p) ÷ n) with p = 0.4226. Calculate it (in percentage points) for n = 100 and n = 10,000.
Answer: n = 100: about 4.94 points. n = 10,000: about 0.49 points.
Working: √(0.4226 × 0.5774 ÷ 100) ≈ 0.0494
4.Change the seed to mylab and run 100 samples again. Is the count the same as before? Why or why not?
Answer: No: a different seed starts a different random sequence, so the count differs. The difference between the two runs is sampling error. Using the same seed repeats a run exactly, which makes experiments checkable.
5.By what factor must the number of samples increase to halve the typical error?
Answer: 4 times as many samples, because the error shrinks with the square root of n.