Approximate a Rare-Event Binomial Probability

Let (X\sim\operatorname{Binomial}(1000,0.001)). Estimate (P(X>5)) without summing 1,001 binomial terms. Derive the approximation, calculate a numerical value, and explain why the chosen limiting distribution is appropriate. ### Constraints & Assumptions - Interpret (X>5) as (X\ge6). - Show the parameter of the approximating distribution. - Retain enough terms to make the small upper-tail probability credible. ### Clarifying Questions to Ask - Is a Poisson approximation acceptable, or is an exact binomial benchmark also requested? - How many significant digits are useful for this rare event? - Are the 1,000 Bernoulli trials independent with a common success probability? ```hint Hold the expected count fixed For large (n) and small (p), compare the binomial count with a Poisson random variable whose mean is (np). ``` ### What a Strong Answer Covers - Selection of (lambda=np=1). - Use of the complement through counts zero to five. - A numerical probability on the correct order of magnitude. - Discussion of the rare-event assumptions and approximation error. ### Follow-up Questions - How would you bound or estimate the Poisson approximation error? - What if the success probabilities differ slightly across trials? - How many trials would be needed for the probability of at least six events to exceed 5 percent when (p=0.001)?