If you send 5 emails, each with a 10% chance of getting a reply, what is the probability of getting at least one reply?

Instruction: Assume five mutually independent reply outcomes with a 10% reply probability for each email. Use the complement rule, then explain why independence matters.

Context: This question assesses the candidate's understanding of the complement rule and their ability to apply it to calculate probability.

Updated

Official answer

If the five reply indicators are mutually independent and each has reply probability 0.1, the chance of no replies is 0.9^5 = 0.59049. Therefore P(at least one reply) = 1−0.9^5 = 0.40951, or 40.951%.

It is the reply outcomes that must be independent, not merely the sending actions. Repeated messages to one recipient or a shared outage can induce dependence. With only five marginal probabilities of 0.1, the probability of any reply can range from 0.1 to 0.5; the specific 40.951% result needs the stronger model.

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