For a Poisson distribution with λ = 3, what is P(X = 0) approximately?

• A. e^{−3} ≈ 0.0498
• B. 1 − e^{−3}
• C. 0
• D. e^{3} ≈ 20.09

Answer: (A)

For a Poisson distribution, the probability of observing zero events is P(X = 0) = e^{−λ} because P(X = k) = e^{−λ} λ^k / k! and λ^0 / 0! equals 1. With λ = 3, this becomes P(X = 0) = e^{−3}.

Numerically, e^{−3} ≈ 1 / e^3, and since e^3 ≈ 20.085, e^{−3} ≈ 0.0498. So the probability of zero events is about 0.0498. The other options don’t fit: 1 − e^{−λ} would be the probability of at least one event (not zero), e^{3} is far greater than 1 and not a probability, and 0 would only occur if λ were 0.