Let's say you have a single-lane road (single in this case meaning single lane in each direction, or what you could also call two-lane). Let's say you have a random number of vehicles on a given 10-mile stretch of road. The speed limit is 45 mph, but some people go faster and some go slower.
Is there any math that describes the likelihood that faster vehicles will overtake slower vehicles at "choke points" (points where oncoming traffic makes passing impossible)? (Assume a straight, flat road where safe passing is always allowed.) EDIT: Choke points are points where oncoming cars make it impossible to pass without the overtaking vehicle having to slow down and wait for the oncoming car (or cars) to pass.
Obviously, the answer has to be some kind of curve, because if there is only one vehicle on the road the probability is 0 that a choke point will be encountered, and if the road is completely filled with vehicles in both directions (or even one) the probability is 1 that a choke point will be encountered at any given time.
Sorry if this is a stupid question. But it seems like there should be a way to express this mathematically.
EDIT 2: Limiting the scope of the problem
Thanks to Mike Spivey, and I hope this helps narrow the problem so that may be in some way answerable. If not, I'll just declare defeat and retire from the field.
Let's say that we have two cars going in one direction: a slow vehicle (Vehicle A) traveling at 30mph and a faster vehicle (Vehicle B) traveling at 60mph. At the start of the problem the A is somewhere between mile 2 and mile 4 on the 10-mile road, and B is at mile 0. Additionally, there will be four oncoming vehicles randomly distributed, and these vehicles are randomly traveling at 30 or 60mph. The overtaking vehicle requires .5 miles of "clear" space to pass safely if the oncoming vehicle is traveling at 60mph, and .25 if it is traveling at 30mph.
Assume further that the car lengths do not matter, but that a "choke point" (or bottleneck, if you prefer) is not reached until B gets within .05 miles of the rear bumper of A, at which point B has to pass A or decelerate. We will call the latter condition a "deceleration event," (DE) and I would like to be able to calculate the probability that at least one DE will occur for Vehicle A along the 10-mile route.
Perhaps this clarification is still insufficient to render the problem solvable, but I appreciate anyone taking the time to consider it.