Suppose that during the daytime, 60% of all vehicles crossing a toll bridge are

Question

Suppose that during the daytime, 60% of all vehicles crossing a toll bridge are passenger vehicles. Assume that successive vehicles crossing the bridge are independent events.

a. What is the probability more than 5 vehicles (non-passenger vehicles) cross the bridge before the first passenger vehicle crosses the bridge?

b. During a 10-minute span, 15 vehicles cross the toll bridge. What is the probability that exactly one-third of these vehicles are not passenger vehicles?

c. During the time span when 15 vehicles cross the toll bridge, what is the expected number of these vehicles that will be passenger cars? What is the standard deviation for the number of passenger cars in these 15 vehicles?

d. Suppose that the toll for passenger cars is $1.00 and $2.50 for other vehicles. If 15 vehicles cross the bridge during a particular time period, what is the mean and standard deviation for the amount of money that will be paid in tolls?

Explanation / Answer

Prob for any vehicle to be passenger vehicle = 0.60

X - no of vehicles passing is independent with two outcomes

Y = non passenger vehicle passing

P(y = 5 and after y x = 1) = 0.45(0.6)

=0.006144

b)  Here n = 15 p = 0.6

P(x = 10) = 0.1859

c) Here n = 15 p = 0.6

E(X) = np = 9

Var(x) = npq = 3.6

Std dev (x) = 1.897

-----------------------------

d) If M is the money collected

M 1.00 2.50

p 0.60 0.40

E(M) for one vehicle = 0.6+1 = 1.6

For 15 vehicles Expected value = 15(1.6) = 24.0

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