Poisson Distribution

To solve the problem, we need to analyze the properties of the Poisson distribution. A random variable X follows a Poisson distribution with parameter λ, where the probability of X taking a value k is given by:

Given that , we have:

Simplifying, we get:

Multiplying both sides by 2 to eliminate the fraction:

Rearranging gives:

Factoring out λ:

This implies or . λ cannot be 0 in a Poisson distribution, so .

Now we need to find :

Calculating :

Thus:

Therefore, the probability is .