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Define a family of error events
Let E_i be a false rejection for test i. In this exercise every null is true, each event has probability a, and the error events are independent. These are stronger assumptions than merely using the same reported significance level in a spreadsheet.
Calculate no errors before taking the complement
The chance of no error for one test is 1 − a. Under independence, multiply that chance m times for no errors across the family. Subtract from one for at least one error.
V is the number of false rejections; all m nulls are true and their rejection events are independent with probability a.
Compare probability with expected count
The expected count is ma by linearity of expectation. It does not require independence when each event probability equals a. A count can exceed one; a probability cannot.
| m | At least one error under independence | Expected count |
|---|---|---|
| 1 | 0.05 | 0.05 |
| 10 | 0.4012631 | 0.5 |
| 20 | 0.6415141 | 1 |
Do not erase dependence or conservative tests
If each test has error probability at most alpha, exact equality need not hold. If error events are dependent, the product step need not hold. A general union bound instead limits the family probability by the sum of the individual probabilities, capped at one.