Question

Traffic on Rosedale Road in Princeton, NJ, follows a Poisson process with rate 6 cars per minute. A deer runs out of the woods and tries to cross the road. It takes the deer a random time (independent of everything else), uniformly distributed between 2 and 5 seconds, to cross the road. If there is a car passing while the deer is on the road, it will hit the deer with portability 2/3 (independent of everything else). Find the probability of a collision.

Answer 1

It is given that, 6 cars are passing through the road per minute.

That is, 6/60= 0.1 of the car will pass through the road per second.

Then for 5 seconds, the rate would be 5*0.1= 0.5.

Consider a random variable X which represents the number of cars in the next 5 seconds. It is given that rate will follows a poisson distribution.

That is,

P(there is a collision)= 1- P(no car coming in next 5 seconds)

= 1- P(X=10)

= 1- 0.6065

= 0.3935.

Chance of collision if the deer only needs 2 seconds to cross the road.

Let Y be another random variable that represents the number of cars passing the road in 2 seconds. It is given that the rate follows a poisson distribution.

That is,

P(there is a collision)= 1- P(no cars coming in next 2 seconds)

= 1-P(Y=0)

= 1- 0.8187

= 0.1813.