Answer to Question #323826 in Statistics and Probability for Bless

Question

 The mean number of automobiles entering a mountain tunnel per two-minute period is one. An excessive number of cars entering the tunnel during a brief period of time produces a hazardous situation. Find the probability that the number of autos entering the tunnel during a two-minute period exceeds three. Does the Poisson model seem reasonable for this problem? 

Answer 1

$X\sim Poiss\left( 1 \right) \\P\left( X>3 \right) =1-P\left( X\leqslant 3 \right) =1-\sum_{i=0}^3{P\left( X=i \right)}=\\=1-\sum_{i=0}^3{\frac{1^ie^{-1}}{i!}}=1-e^{-1}\left( 1+\frac{1}{1}+\frac{1}{2}+\frac{1}{6} \right) =0.0189882\\The\,\,Poisson\,\,model\,\,seems\,\,reasonable\\$