What Is the Probability That 4 or More Recover if a Drug Has No Effect?

[bartowski]

The probability that an individual recovers from an illness in a one-week time period without treatment is 0.1.
Suppose that 20 independent individuals suffering from this illness are treated with a drug and 4 recover in a one-week time period. If the drug has no effect, what is the probability that 4 or more people recover in a one-week time period?

 

[SW VandeCarr]

The probability that an individual recovers from an illness in a one-week time period without treatment is 0.1.
Suppose that 20 independent individuals suffering from this illness are treated with a drug and 4 recover in a one-week time period. If the drug has no effect, what is the probability that 4 or more people recover in a one-week time period?

You're saying the probability of recovery without treatment is 0.1. This means 2 would be expected to recover out of 20 on average. In fact at least 4 people recovered with the drug. To be able to infer the drug had an effect, you really need a placebo treated control group for a proper scientific assessment.

You could possibly use the Poisson distribution to obtain the probability of 4 or more recoveries given the mean is 2. But I'm not sure of the usefulness of such an analysis. Is this one parameter distribution the correct one for your question, or do you need specific information on the variance? In any case, you can't properly assess the drug's effect without a control.