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:

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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 $1-\sum_{k=0}^4 \frac{e^{-4}4^k}{k!}$


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