Question

Suppose that the bus of the sultana has a capacity of 8 passengers and needs At least 5 passengers arrive to make the trip from Mayagüez to San Juan. The clients They arrive on average at the rate of 1 customer every 5 minutes.

a) What is the probability that the bus will go to San Juan in 30 minutes. (suggestion you can use one of the following distributions: the Exponential or the Poisson).

b) What is the probability that more than 2 clients arrive in 20 minutes.

Answer 1

a)

expected number of customer in 30 minutes =30/5 =6

therefore  probability that the bus will go to San Juan in 30 minutes =P(at least 5 passenger arrive)

=P(X>=5)=1-P(X<=4)

=1-(P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4))

=1-(e^-66⁰/0!+e^-66¹/1!+e^-66²/2!+e^-66³/3!+e^-66⁴/4!)

=0.7149

b)expected number of customer in 20 minutes =20/5 =4

probability that more than 2 clients arrive in 20 minutes =P(X>2)

=1-P(X<=2)

=1-(P(X=0)+P(X=1)+P(X=2))

=1-(e^-44⁰/0!+e^-44¹/1!+e^-44²/2!)

=0.9380