Poisson distribution problem - traffic problem Hi So I have this question below. I know I need to model each lane as a separate Poisson distribution. The possible answers are:
a) 11.4%; 22.4%; 33.4%; 44.4%; 55.4%
b) 2.74%; 4.74%; 12.74%; 34.74%; 64.74% 
but obviously I need to know how to get to the answer.

Question:
Vehicles arrive at a rate of $1200$ vehicles per hour at a traffic
  signal. $15\%$ of the vehicles turn right while $85\%$ travel straight
  through. The street has one lane per direction, which is widened at
  the intersection to one straight through lane, and one right-turn
  lane. The right-turn lane can accommodate $5$ vehicles but will be
  blocked if $5$ or more straight-through vehicles are waiting at the
  intersection. Determine the following:
a) The probability that no vehicles are waiting in the right-turn lane
  while there is a total of $5$ vehicles waiting at the intersection. 
b)The probability that more than $5$ right-turn vehicles will arrive at
  the intersection during the next red period of $45$ seconds.

For question a) I think I need to find $\Pr(X=5)$ for the straight lane while $\Pr(X=0)$ for the right lane. 
$$
\Large P_P(x; qt) = \frac{(q\cdot t)^x\cdot e^{-q\cdot t}}{x!}
$$
Where I am having trouble though is how do I determine $q$?
 A: It's tricky to answer homework problems, but one indirect way of looking into part (b) of the problem is to simulate (the closed-form solution still up to you). It seems as though your question elicited quite a bit of interest, and it could possibly get more entries if you erased the actual answers. So here it goes...
In R:
set.seed(0)
lambda_right = 0.15 * 45 * 1200/60^2        # Rate parameter of 2.25 cars/period (45 sec.)
nsim = 10^6                                 # One million simulations
right_arrivals = rpois(nsim, lambda_right)  # Poisson process rate lambda
mean(right_arrivals > 5)                    # Percentage with more than 5 R turning cars
[1] 0.027423

Here is the plot:

The mean value is clearly consistent with the setup on the simulation ($\lambda =2.25$), and it is clear that the majority of arrivals are below $5$ cars/period, explaining the low percentage for $>5$.
A: Take 2:
a)Conditional probability of 5 cars going straight, given that there are 5 cars already.

=44.37%
==========================================================================
b)

=2.74%
Can anyone confirm my answers?
