Poisson: bad reaction among 2000 (a, b)

n = 2000 is large and p = 0.001 is small, so use the Poisson approximation to the binomial with

λ = np = 2000 × 0.001 = 2, P(X = x) = e^(−λ) λˣ / x! = e⁻² 2ˣ / x!, with e⁻² = 0.135335.

a) Exactly 3 suffer a bad reaction

P(X = 3) = e⁻² · 2³ / 3! = 0.135335 × 8 / 6 = 0.1804

b) More than 2 suffer a bad reaction

P(X > 2) = 1 − [P(0) + P(1) + P(2)]

P(0) = e⁻²            = 0.1353
P(1) = e⁻² · 2        = 0.2707
P(2) = e⁻² · 2²/2     = 0.2707
Sum                   = 0.6767

P(X > 2) = 1 − 0.6767 = 0.3233

Answer: (a) 0.1804, (b) 0.3233