There are 2 entrances to a parking lot Cars arrive at entran

There are 2 entrances to a parking lot. Cars arrive at entrance I according to a Poisson
distribution at an average of 3 cars per hour and at entrance II according to a Poisson distribution at anaverage of 4 cars per hour. What is the probability that 3 cars arrive at the parking lot in a given hour?(Assume that the number of cars arriving at the two entrances are independent.)

Solution

Let X₁ and X₂ be two random variables denoting the number of cars arriving in the parking lot\'s entrance I and entrance 2 respectively in a given hour .

Now according to question ,

X₁~P(3) and X₂~P(4)

So, we have to find the probability that 3 cars arrive at the parking lot in the given hour

It can happen in 4 ways i.e

0 cars in entrance I and 3 cars in entrance II

3 cars in entrance I and 0 cars in entrance II

1 cars in entrance I and 2 cars in entrance II

2 cars in entrance I and 1 cars in entrance II

Hence we have to find,

P(X₁=0, X₂=3)+P(X₁=3, X₂=0)+P(X₁=1, X₂=2)+P(X₁=2, X₂=1)

= 0.0521293 (Simply calculating the respective probailities in Minitab or R)

( Note that as X₁ X₂ are independent hence P(X₁=0, X₂=3)=P(X₁=0)*P(X₂=3) and similarly others)