A basketball player says she makes 70% of her free throws, then makes only 9 of her next 20. Is she wrong, or was that just a bad day? In Grade 7 you used simulations to estimate probabilities. Here we use them to judge a claim, comparing what happened with what the claimed model usually produces. It is the same long-run thinking as expected value.
Key idea
To check whether data are consistent with a model: (1) simulate the process many times assuming the model is true; (2) see how often the simulation gives a result at least as extreme as the real one. If that almost never happens, the data are evidence against the model. If it happens often, the data are consistent with the model (which is not the same as proving it).
Why it works
Chance alone makes results vary. A 70% shooter will not make exactly 14 of 20 every time; sometimes 12, sometimes 17. The simulation shows the range that chance produces when the claim is true. A result far outside that range would be a strange coincidence if the claim were true, so it casts doubt on the claim. This is the logic behind every significance test later in the course.
Worked example: A doubtful claim
A player claims to make 70% of free throws, then makes 9 of 20. In 100 simulated sets of 20 shots by a 70% shooter, only 2 sets had 9 or fewer makes. What do you conclude?
- The simulation assumes the claim is true and shows what chance produces.
- A result as low as 9 happened in only 2 of 100 runs: an estimated probability of about $0.02$.
- That would be unusual for a 70% shooter, so the data give evidence that her rate is lower (or that something changed).
Answer: Evidence against the 70% claim.
Worked example: A result chance explains
A coin lands heads 12 times in 20 flips. In 100 simulations of a fair coin, 25 runs gave 12 or more heads. Is the coin unfair?
- About 25% of fair-coin runs are at least this extreme.
- A result that happens a quarter of the time is not surprising.
- The data are consistent with a fair coin. They do not prove the coin is fair; they just give no real evidence against it.
Answer: No evidence that the coin is unfair.
Worked example: Designing a simulation
Describe how to simulate one set of 20 free throws by a 70% shooter with a random number generator.
- Let each random digit 0 to 9 be one shot: digits 0–6 (7 of the 10 digits) mean "made," 7–9 mean "missed."
- Generate 20 digits and count the makes. That is one run.
- Repeat for many runs (100 or more) and record the count each time.
Answer: Use digits 0–6 as makes, 20 digits per run, many runs.
Common mistake
Concluding that a model is proven true because the data are consistent with it. "12 heads in 20 is not unusual for a fair coin" does not prove the coin is fair: a slightly unfair coin would often give 12 heads too. Check your wording: "consistent with" or "no evidence against," never "proves."