An average of 2 cars cross the Mackinac Bridge toll plaza ev

An average of 2 cars cross the Mackinac Bridge toll plaza every 5 minutes. What is the probability that more than 3 cars cross the toll plaza between 1:00 pm and 1:10 pm

Solution

We have given that an average of two cars cross the Machinac bridge toll plaza every 5 minutes.

We have to find the probability that more than 3 cars cross the toll plaza between 1.00 pm to 1.30 pm.

For every 5 minutes on an average 2 cars cross the bridge then for every 10 minutes on an average 2*2 cars cross the bridge.

Let X be a random variable that number of cars cross the toll plaza.

X ~ poison(mean =4)

We have to find the probability that,

P(X > 3) = 1 - P(X3) = 1 - [ P(X=0) + P(X=1) + P(X=2) + P(X = 3) ]

The pmf pf Poison distribution is,

P(X = x) = e^-4 4^x / x!

Now we have to find Probabilities for x=0,1,2 and 3.

P(X=0) =  e^-4 4⁰  / 0! = 0.0183

P(X=1) = e^-4 4¹ / 1! = 0.0733

P(X=2) = e^-4 4² / 2! = 0.1465

P(X=3) = e^-4 4³ / 3! = 0.1954

P(X > 3) = 1 - [ 0.0183 + 0.0733 + 0.1465 + 0.1954 ]

P(X > 3) = 0.5665