probability of negative binomial distribution

Question

I found this question on the internet:

Police conduct random breath tests on drivers during a busy night in the city. 3% of drivers drink and drive at the time. $X$ is is the number of drivers that police need to test to get the first case of drinking and driving. $Y$ is the number of drivers tested to get 3 such cases.

Since $Y$ is a negative binomial distribution, how would you be able to calculate $P(Y > 40)$?

I understand that the long way to do this would be:

$P(Y > 40) = 1 - P(Y=40) - P(Y=39) - P(Y=38)....$

Is their an easier way to do this?

Answer 1

Another way to do this would be to say that to get $Y > 40$, you would need that among the first $40$ drivers, $2$ or fewer were drinking.

Let $Z$ be the number of drunk drivers among the first $40$ drivers, and just calculate $P(Z=0) + P(Z=1) + P(Z=2)$.

These probabilities can all be found using a simple binomial distribution, and this way you only need to do three calculations instead of $40$.