Autos arrive at a toll plaza located at the entrance to a bridge at a rate of 25

Question

Autos arrive at a toll plaza located at the entrance to a bridge at a rate of 25 per minute during the 5:00-to-6:00 P.M. hour. Determine the following probabilities assuming that an auto has just arrived.

a. What is the probability that the next auto will arrive within 66 seconds (0.1minute)?

b. What is the probability that the next auto will arrive within 22 seconds (0.0333minute)?

c. What are the answers to (a) and (b) if the rate of arrival of autos is 60 per minute?

d. What are the answers to (a) and (b) if the rate of arrival of autos is 30 per minute?

Explanation / Answer

Autos arrival rate = 25 per minute = 25/60 per second

(a) Here expected number of that will arrive in 6 seconds = 25/60 * 6 = 2.5 autos

so Pr(Next auto will arrive in 6 seconds) = 1 - Pr(no auto will arrive in 6 seconds)

= 1- POISSON (X = 0; 2.5)

= 1- e-2.5 = 1 - 0.0821 = 0.9179

(b) expected number of autos to be arrive in next 2 seconds = (25/60) * 2= 5/6 auto

so Pr(Next auto will arrive in 2 seconds) = 1 - Pr(no auto will arrive in 2 seconds)

= 1- POISSON (X = 0; 5/6)

= 1- e-5/6 = 1 - 0.4346 = 0.5654

(c) Here arrival rate of autos =60 per minute

Here expected number of that will arrive in 6 seconds = 60/60 * 6 = 6 autos

so Pr(Next auto will arrive in 6 seconds) = 1 - Pr(no auto will arrive in 6 seconds)

= 1- POISSON (X = 0; 6)

= 1- e-6 = 1 - 0.00248 = 0.99752

expected number of autos to be arrive in next 2 seconds = (60/60) * 2= 2 auto

so Pr(Next auto will arrive in 2 seconds) = 1 - Pr(no auto will arrive in 2 seconds)

= 1- POISSON (X = 0; 2)

= 1- e-2 = 1 - 0.1353 = 0.8647

(d)

Here arrival rate of autos =30 per minute

Here expected number of that will arrive in 6 seconds = 30/60 * 6 = 3autos

so Pr(Next auto will arrive in 6 seconds) = 1 - Pr(no auto will arrive in 6 seconds)

= 1- POISSON (X = 0; 3)

= 1- e-3 = 1 - 0.0498 = 0.9502

expected number of autos to be arrive in next 2 seconds = (30/60) * 2= 1 auto

so Pr(Next auto will arrive in 2 seconds) = 1 - Pr(no auto will arrive in 2 seconds)

= 1- POISSON (X = 0; 1)

= 1- e-1 = 1 - 0.3679 = 0.6321

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