Question

A personnel from the Traffic Engineering Center was assigned to observe a particular intersection. Historical data on the characteristics of vehicles passing thru this intersection revealed the following data:

                        Probability that a vehicle is a smoke belcher = 0.30

                        Probability that the vehicle does not have a license plate = 0.002

                        Probability that the vehicle passing is a truck = 0.10

                        Average number of vehicles passing per minute = 10

1. What is the probability that the tenth vehicle is the fourth truck that passed the intersection?

1. 0.0045    c. 0.0254   e. 0.2138
2. 0.1172    d. 0.0444

2. What is the probability that the first smoke belcher is the seventh vehicle that passed this intersection?

1. 0.5333   c. 0.3555   e. 0.3053
2. 0.0553   d. 0.0353

3. What is the probability that twelve vehicles will pass the intersection from 4:00 to 4:02 pm ?

1. 0.0054   c. 0.0176   e. 0.0267
2. 0.9524   d. 0.9733

Answer 1

1. What is the probability that the tenth vehicle is the fourth truck that passed the intersection?

Let the probability that the vehicle passing is a truck = p=0.10

For the 10th vehicle passing the intersection to be the 4th truck, we know that out of the 9 vehicles that passed the intersection before this there would be 3 trucks

Let X be the number of trucks out of 9 vehicles that pass the intersection. We can say that X has Binomial distribution with parameters, number of trials (number of vehicles passing the intersection) n=9 and the success probability (the probability that the vehicle passing is a truck) p=0.10

the probability that X=3 trucks passed the intersection is

Next the 10th vehicle that passes needs to a truck. The probability of that happening is 0.10

the probability that the tenth vehicle is the fourth truck that passed the intersection is

ans: a. 0.0045

2. What is the probability that the first smoke belcher is the seventh vehicle that passed this intersection?

Probability that a vehicle is a smoke belcher = 0.30

Probability that a vehicle is not a smoke belcher = 1-0.30=0.70

For the 7th vehicle to be the first smoke belcher, we can say that the previous 6 vehicles would have to be not smoke belchers.

The probability that the first smoke belcher is the seventh vehicle that passed this intersection is

ans: d. 0.0353

3. What is the probability that twelve vehicles will pass the intersection from 4:00 to 4:02 pm ?

Average number of vehicles passing per minute = 10

Let N(t) be the number of vehicles that pass through the intersection in t minutes. We can say that N(t) has a Poisson distribution with rate

That is the probability that N(t)=n vehicles pass through the intersection in t minutes is given by

Here, we need the number of vehicles that pass though the intersection between 4:00 to 4:02 PM, which is a time interval t=2 minutes

That is the probability that N(2)=n vehicles pass through the intersection in 2 minutes is given by

the probability that twelve vehicles will pass the intersection from 4:00 to 4:02 pm is

ans: c. 0.0176