Relationship between Poisson and Exponential distribution, automobiles arrive per minute

Question

I am struggling to understand the Poisson and Exponential distributions.

The number of automobiles that arrive at a certain intersection per minute has a Poisson distribution with a mean of 5. Interest centers around the time that elapses before 10 automobiles appear at the intersection.

What is the probability that more than 10 automobiles appear at the intersection during any given minute of time?

What is the probability that more than 2 minutes elapse before 10 cars arrive?

I have the formula P(x,λt) = (e^(-λt)*(λt)^x)/x! and f(x) = λe^(-λx) where λ=5/minute, t is the time, and x is the number of automobiles. I tried integrating f(x)dx from 10 to infinity but got the nonsense answer of e^(-50).

Please explain the Poisson distribution and how to apply it to this problem.

Answer 1

For the first problem, it is easier to first find the probability of the complement, that is, the probability that there are $10$ or fewer cars in $1$ minute. The probability that exactly $k$ cars arrive in one minute is $e^{-5}\frac{5^k}{k!}$. Now you can put the pieces together.

For the second problem, use the fact that the number $Y$ of cars in $2$ minutes has Poisson distribution, mean (parameter) $(2)(5)$. We want the probability that more than $2$ minutes elapse before $10$ cars arrive. This is the probability that there are fewer than $10$ cars in $2$ minutes, that is, $\Pr(Y\le 9)$.