Question

Cars cross a certain point on the highway in accordance with a Poisson process with rate = 3 per minute. If Al runs across the highway at some random time, what is the probability that he will avoid being hit by a car if the amount of time that it takes him to cross the road is 10 seconds? Hint: Assume that if a car goes by while Al is on the highway, that it will hit him.

Answer 1

Answer:

The probability is 0.2212

Solution:

As per the question:

Poisson rate, $\lambda = 3/min = \frac{3}{60}\ s = 0.05\ s$

Now,

Let

X: time of waiting for the next vehicle on the highway

Now,

To find the probability of the next vehicle to arrive after 10 s

The probability distribution function is given by:

$f(x) = \lambda e^{- \lambda x}$

Now,

P(X < x) = $1 - e^{- \lambda x}$

$P(X \leq x) = 1 - e^{- 0.05 x}$

For X> 0,

P(X > x) = $e^{- \lambda x}$

P(X < 5) = $1 - e^{- 5\lambda} = 1 - e^{- 0.25} = 1 - 0.7788 = 0.2212$