I am trying to solve the following question.
A pedestrian needs at least 8 seconds to cross a dual-lane carriage-way. Vehicles in the near-side lane arrive at the rate of 10 per minute and vehicles in the far-side lane at the rate of 8 per minute. Making suitable assumptions:What is the probability that the nearest vehicle (in time) will be in the near-side lane?
Thoughts towards solution
We have been given two Poisson processes the first with mean $\lambda_1 = 10 $ cars per minute and the second with $\lambda_2 = 8 $ cars per minute. The interval between two successive arrivals has a negative exponential distribution in each case with means $\dfrac {1}{\lambda_1}$ and $\dfrac {1}{\lambda_2}$. I think if i understand the problem correctly we are interested in computing
Let $T_1$ be the arrival time of the next vehicle in the near-side lane. Let $T_2$ be the arrival time of the next vehicle in the far-side lane.
$P(T_1
This evaluated to 0.0694 which seems intuitively kind of low. I suspect that i have made a mistake. Any help would be much appreciated.
Thanks in advance.