The mean number of automobiles entering a mountain tunnel pe

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. a) What is the probability distribution of the number of autos entering the tunnel? b) Find the probability that the number of autos entering the tunnel during a two-minute period exceeds three

Solution

a)

This is a Poisson distribution, as the number of occurences per unit time (2 minutes) is constant (1 car).

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b)

Note that P(more than x) = 1 - P(at most x).          
          
Using a cumulative poisson distribution table or technology, matching          
          
u = the mean number of successes =    1      
          
x = our critical value of successes =    3      
          
Then the cumulative probability of P(at most x) from a table/technology is          
          
P(at most   3   ) =    0.981011843
          
Thus, the probability of at least   4   successes is  
          
P(more than   3   ) =    0.018988157 [ANSWER]