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State a complete synthetic experiment
Imagine ten true null hypotheses and ten independent error events, each with rejection probability 0.05. This experiment is designed for arithmetic clarity; ten signals tested on the same return history need not have independent errors.
Calculate the expected number
Write the count as the sum of ten indicators, one per false rejection. Each indicator has expectation 0.05, so the total expectation is 10 × 0.05 = 0.5. An expectation of half an error is an average over repeated experiments, not a possible half-rejection in one experiment.
Calculate at least one error
The no-error probability is 0.95^10 ≈ 0.5987369. Its complement is approximately 0.4012631. The outcome V = 0 has positive probability, while other experiments can contain more than one error.
| Question | Result |
|---|---|
| How many false rejections on average? | 0.5 |
| Chance of at least one false rejection? | About 40.1263% |
| Chance of no false rejections? | About 59.8737% |
Keep the stated population of nulls visible
When some null hypotheses are false, total rejections include potential true discoveries and false rejections. Counting every rejection as an error then becomes incorrect. Record which mathematical scenario is being discussed before applying the all-true-null illustration.