$X_t$ is the amount of transmitted particles in a specific time unit, described by $$\pi(X_t=k)=\frac{\lambda^k}{k!}\cdot e^{-\lambda}$$
$p$ is the probability that a transmitted particle is monitored. $A_k$ is the event that exactly $k$ particles are transmitted. $B_i$ is the event that exactly $i$ particles are monitored.
Specify the conditional probabilities $P(B_i \mid A_k)$ and calculate $P(B_i \cap A_k)$
I think the formula in question is poisson distribution? Because I find on internet the same formula but it have no $\pi$. Is it important or is only part of name?
Anyway I think conditional probability of it is just $$P(B_i \mid A_k)=\frac{P(B_i \cap A_k)}{P(A_k)}$$
Now need calculate $P(B_i \cap A_k)$ also. How do it good?
I think because the events $A_k$ and $B_i$ are dependent we need use other way, we need use the conditional probability also.
We get $$P(B_i \cap A_k) = P(B_i) \cdot P(A_k \mid B_i) = P(B_i \mid A_k) \cdot P(A_k)$$
Is this solutions good? No sure because all confused notation and question..