Given one lane for turning left and one lane for thru traffic. Cars arrive according to a Poisson process with rate $\lambda = 20$ per minute. An arriving car turns left with ${40\%}$ probability, and wants to go straight with 60% probability. The left-hand turn lane has a max capacity of $10$ cars. If more than $10$ cars arrive when the light is red ($1$ minute long), then the cars wanting to turn left block the progress of cars wanting to go straight.
a. During the $1$-minute period when the light is red, what is the probability that the left-hand turn lane blocks progress of cars wanting to go straight?
b. Given that $11$ left-turning cars have arrived during the $1$-minute period when the light is red, what is the probability that no through-cars are blocked during this period?
My thought (a) is simple. We could denote $X =$ number of cars arrive on the left-hand turn lane during red line where $X\sim $ PP$(20\times 0.4 = 8)$. Thus $P(X\geq 11) = 1-P(X=0) - P(X=1) -\cdots - P(X=10) = 1- e^{-8} - e^{-8}*8 - \cdots e^{-8}\frac{8^{10}}{10!} = 0.112.$
(b) I could only see that since $11$ left-turning cars have arrived already, there are no through cars blocked if and only if there are no cars arriving going straight during that $1$-minute period. Thus, $P(N(1) = 0) = e^{-12}$.
My question I'm not sure if my thought for part (b) is correct, since the problem seems to be tricky. Could anyone please help review my solution above and give some thoughts? Would really appreciate any input.