Independent solution

How to solve this Poisson Distribution question

Setup

Setup

Let N be the relevant monthly count. Its Poisson mean is 4, and the available information restricts the outcome to N at least 1.

N∼Poisson⁡(4)N\sim\operatorname{Poisson}(4)
Pr⁡(N=0)=e−4\Pr(N=0)=e^{-4}

Model

Model

Because the target event is contained in the observed event, use a ratio of its unconditional probability to the probability of a nonzero count.

Pr⁡(N>3∣N≥1)=Pr⁡(N>3)Pr⁡(N≥1)\Pr(N>3\mid N\ge 1)=\frac{\Pr(N>3)}{\Pr(N\ge 1)}

Compute

Compute

Evaluate the Poisson lower-tail complement in the numerator and remove the zero mass in the denominator.

Pr⁡(N>3)=1−e−4(1+4+422!+433!)=0.5665298796…\Pr(N>3)=1-e^{-4}\left(1+4+\frac{4^2}{2!}+\frac{4^3}{3!}\right)=0.5665298796\ldots
Pr⁡(N≥1)=1−e−4=0.9816843611…\Pr(N\ge1)=1-e^{-4}=0.9816843611\ldots
Pr⁡(N>3∣N≥1)=0.5665298796…0.9816843611…=0.5770998318…\Pr(N>3\mid N\ge1)=\frac{0.5665298796\ldots}{0.9816843611\ldots}=0.5770998318\ldots

Answer

Answer

The requested conditional probability rounds to 0.577.

0.577(B)\boxed{0.577\quad\text{(B)}}