For a Poisson distribution with λ = 3, what is P(X ≥ 3) approximately?

For a Poisson distribution with λ = 3, what is P(X ≥ 3) approximately?

• A. ≈ 0.5768
• B. ≈ 0.4232
• C. ≈ 0.0005
• D. ≈ 0.9995

Answer: (A)

For a Poisson distribution with parameter λ, P(X ≥ k) is 1 minus the sum of the probabilities up to k−1: P(X ≥ 3) = 1 − [P(0) + P(1) + P(2)]. Compute with λ = 3: P(0) = e^{-3} ≈ 0.049787 P(1) = 3 e^{-3} ≈ 0.149361 P(2) = 9/2 e^{-3} ≈ 0.224041 Sum ≈ 0.423189, so P(X ≥ 3) ≈ 1 − 0.423189 ≈ 0.576811, about 0.5768. So the probability of at least 3 events is roughly 0.5768, matching the given value.

For a Poisson distribution with parameter λ, P(X ≥ k) is 1 minus the sum of the probabilities up to k−1: P(X ≥ 3) = 1 − [P(0) + P(1) + P(2)]. Compute with λ = 3:

P(0) = e^{-3} ≈ 0.049787

P(1) = 3 e^{-3} ≈ 0.149361

P(2) = 9/2 e^{-3} ≈ 0.224041

Sum ≈ 0.423189, so P(X ≥ 3) ≈ 1 − 0.423189 ≈ 0.576811, about 0.5768.

So the probability of at least 3 events is roughly 0.5768, matching the given value.