I have a question like this:

A missile protection system is set up in a particular zone. The system consists of $ n$ radar sets operating independently. Each set has a probability of $ 0.95$ of detecting a missile which enters the zone.

Question

1. Suppose that there are 6 radar sets operating in a particular day (i.e. $ n = 6$ ), Given that a missile is detected by at least one set, what is the conditional probability that it is only detected by exactly one set?

We have: $ X \sim B(6, 0.95)$ . Therefore: $ $ P(X = 1 | X \geq 1) = \frac{P(X = 1)}{P(X \geq 1)} = \frac{2}{1122807} \approx 0$ $

2. If the probability of detecting a missile in the zone is required to be at least $ 0.9999$ , what is the smallest $ n$ can be?

Assuming: $ X \sim B(n, 0.95)$ . We need: $ P(X \geq 1) \geq 0.9999$ or equivalently: $ $ P(X = 0) \leq 0.0001 = \binom{n}{0}*0.95^{0}*0.05^{n} = 0.05^{n}$ $ Therefore, $ n \geq 3.0744 $ . Minimum $ n$ is $ 4$ .

Is my solution right?