Poisson distribution: trucks and cars

Question

This is some probability problem that I conjured up. Can anyone check whether this problem makes sense and has a solution?

Assume that the traffic on Spooner street follows a Poisson process with a rate 2/3's of a vehicle per hour. 10% of the vehicles are trucks, the other 90% are cars.

Let's say 100 vehicles passed by in Spooner street. In this case, I'm interested in the case where each truck is separated by at least 4 cars. For example:

'yes' event: car truck car car car car car truck car -> there are 5 cars between a pair of trucks
'no' event: truck car car truck only two cars between a pair of trucks

What is the probability that these 100 vehicles satisfy the property of having each pair of trucks separated by at least 4 cars?

Are there any flats for rent on Spooner Street? $2/3$ of a vehicle per hour is a traffic rate I could live with :-) — joriki, Mar 02 '13 at 00:26
People say Spooner street is somewhere in Quahog, Rhode Island. But I'm afraid you'll not like the Griffins in the neighborhood. — Federico Magallanez, Mar 02 '13 at 00:41

score 1 · Accepted Answer · answered Mar 02 '13 at 00:23

1

No, it doesn't make sense, in the sense that the Poisson process doesn't enter into it. Once you know that there were $100$ vehicles, it becomes equivalent to $100$ unfair coin tosses that can come up trucks or cars, and you might as well not mention the Poisson process (in particular its rate) in the first place (unless you made this up to confuse your students :-).

answered Mar 02 '13 at 00:23

joriki

238,052

Wow! Thank you for your quick answer. I agree with you. – Federico Magallanez Mar 02 '13 at 00:37

score 1 · Answer 2 · answered Mar 02 '13 at 00:26

1

I agree with joriki, once you know they are 100 vehicles, then all you need to do is some combinatorial counting on how many ways you can order them given that the distance between each two truck is at least 4 cars.

answered Mar 02 '13 at 00:26

Mohamad

259

Poisson distribution: trucks and cars

2 Answers2