Probability problem with patrol cars

Reply Tue 5 Dec, 2017 12:20 pm
Two patrol cars are located at points A and B of a motorway, which are 80 km apart. If an incident happens, the nearest patrol car goes there . What is the probability that the patrol car will need to travel more than 30 km?

I need some help. I cannot recognize the distribution of the problem!
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Reply Tue 5 Dec, 2017 03:20 pm
Assuming cars are stationed every 80 km, each car will repond to incidents up to 40 km on either side of it. So each responds over a range of 80km, of which 20 km (10 each side). is more than 30 km from its station.
So p=20/80=1/4.

However, since only 2 cars are mentioned, the problem seems insoluble since there is a theoretically uncountable number of inciden throughout the range of the network outside the 2 vehicles stations.
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Reply Tue 5 Dec, 2017 04:37 pm
As mentioned above, the probability of patrolling outside the line segment AB hasn't been stated, and so that scenario isn't the question being asked. So we can deduce the question being asked is where there is enough information present to solve. (If not, then hire another teacher)

Maximum distance of a patrol is half the line segment between them, so AB/2 km. The patrol distance between A and B is distance = (B - A)/2 = 40 km. If accidents occur uniformly across that stretch of road, then it should be that each patrolman gets half the accidents and 1/4 of the accidents result in police responding from 30+ km.

Next question: Did accidents occur at a significantly different rate in any segment of those territories of responsibility?
Correct response: Tighten patrol region size until the probability (of desirable response time) is the same across each region.
Exercise: Describe how accidents effect patrol positioning
Reply Tue 5 Dec, 2017 10:46 pm
Most accidents happen within 5 km of home so what you should do is not travel there on a uniform basis.
Reply Wed 6 Dec, 2017 02:17 am
Smile Correct !....and don't bet on the National Lottery because the chances of being killed on the way to buying the ticket are far in excess of winning !
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Reply Wed 6 Dec, 2017 05:31 pm
So there's network of patrol points that would form the smallest steps necessary to reallocate troops to meet maximum allowed response time. The set of active patrol points are a small subset of the underlaying grid.

The density of the grid and the math required to update patrol points in response to incoming crime history suggest a software solution would be the best approach.

Historic crime locations are updated, forming a new web of strength patterns that define which patrol points will make each zone meet response specifications. The software then displays those points or relays them through GPS to those on duty.

Eventully a Lieutenant's eyes could read crime incident rate from response team deployments.
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