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If the illness is infectious then you cannot use the binomial distribution because the incidences of illness are no longer independent events, so that the assumptions required for the binomial distribution are not satisfied.

Suppose the illness is not infectious and the "normal" rate of illnesses is p. Then in a group of size n, the number of units suffering has a B(n, p) distribution. You can then determine a critical region at an appropriate level of significance and test the number of victims against that.

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Q: How do you use binomial probability to assess if an outbreak of an illness has occurred or not?
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