Automobiles arrive at a vehicle equipment inspection station according to a Poisson process with a rate of $ \lambda $ = 10 per hour. Suppose that with probability 0.5 an arriving vehicle will have no equipment violations.

What is the probability that exactly 5 have no violations?

I wanted to know if this process was correct?
And I'm a bit confused as to how the probability comes into play here.

$ p\_X(k) = e^{-\lambda} \cdot \frac{\lambda^k}{k!} $

Then, $~ p\_X(5) = e^{-10} \cdot \frac{10^5}{5!} = 0.0378 = 3.78\%$