# Dart board probability using line method with Poisson application

You randomly throw darts at a dartboard, one dart every second. Suppose that every dart independently hits the dartboard at distance X from the center, where X is a Unif[0,30] random variable. Your target, the bullseye, is located around the center and has radius 2.

Suppose you throw darts for 1 minute. Approximate the probability that you score more than 5 bullseye.

the solutions given are, once using Bin and then using poisson in this manner:

Poisson(4) - RV Y
P(Y>5) = 1- $$\frac{643}{15} e^{-4}$$

The Poisson : Poiss($$\lambda$$) = $$\frac{\lambda^k}{k!}e^{-\lambda}$$

so in their solution they are stating $$\lambda = 4$$ and K is supposed to be k=60 , no?

because $$\frac{4^{60}}{60!}$$ doesn't equal to $$\frac{643}{15}$$

Is it a typo or am I missing something here?

For the Poisson one, we are finding the probability of $$Y>5$$. We take away the probability of $$Y\leq 4$$. Overall this is $$1-\sum_{k=0}^4 \frac{e^{-4}4^k}{k!}$$