Question

This problem is based on poisson process Cars arrive at a rate alpha-500/HOUR A pedestrian arrives at a crossing point right after the event of a car passing by. If the pedestrian needs 5 seconds to cross the street, what is the probability that he/she will cross safely without getting hit by a car?

Answer 1

Car Arrival rate N follows poisson distribution with $\lambda$ = 500/hour = 0.139/sec
PDF of Poisson Distribution:
$P(N(t) = n) = e^{-\lambda t}\frac{(\lambda t)^{n}}{n!}$
Probability that a pedestrian safely crosses the street = Probability that no car arrives during the next 5 seconds = Probability that number of car arrivals n in time t=5 seconds is n=0
$P(N(t=5) = 0) = e^{-\lambda *5}\frac{(\lambda *5)^{0}}{0!}$