A monitoring script records the number of checkout failures per minute on a high-traffic e-commerce platform. Historical data indicate that failures occur independently at a constant average rate of 2.3 per minute. Assuming this process follows a Poisson distribution, which of the following values is closest to the probability that at least five checkout failures will be observed in a randomly selected minute?
For a Poisson(λ = 2.3) random variable X, the probability of exactly k events is P(X = k) = e^{-λ} λ^ ⁄ k!. The required probability is P(X ≥ 5) = 1 − P(X ≤ 4). Computing the cumulative probability up to k = 4: P(0) ≈ 0.1003, P(1) ≈ 0.2306, P(2) ≈ 0.2652, P(3) ≈ 0.2033, P(4) ≈ 0.1169. The sum is about 0.9163. Subtracting from 1 gives P(X ≥ 5) ≈ 0.084. The listed value closest to this result is 0.084. The other choices correspond to unrelated cumulative or single-point probabilities (e.g., exactly four events, at least four, or at most four) and therefore do not answer the question.
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