We have all seen the recent police crackdown on speeding, and the stricter enforcement whereby they are likely to stop anyone travelling even 1 kmh over the limit and ticket anyone travelling at more than 4 kmh over the limit. We have also seen the police claim credit anytime there is a reduction in the road toll and blame anyone or anything else when there is an increase in the road toll.
We have all seen the average road speeds fall, and become erratic, with more drivers travelling so far below the speed limit that they hold up other traffic, and then increase their speed at passing lanes so few others can pass. We have also noticed that even experienced drivers are spending more time looking at their speedometers and less time looking at the road or observing road conditions, other traffic and additional road hazards.
In those countries where the speed limit has been increased, or police focus has reduced, there has been a reduction in the road toll, while those countries that have increased speed limits or reduced police focus on speeding have seen increases in road toll, and New Zealand’s experience is consistent with those overseas observations. The places that have seen the greatest reduction in road toll are those that have set the speed limits at the speed that 85% of drivers would stay below “if there was no limit set”. In these situations it has been explained that drivers are less stressed and more able to drive to conditions rather than to an arbitrary speed. Studies in mid-USA where the speed limit was raised showed that the average speed did not increase as much as the increase in the speed limit, showing that drivers were able to set their own speeds where they felt comfortable for the road conditions and for the vehicle they were in.
However, on the TV news a few nights back, Inspector Carey Griffiths stated that, with the stricter speed enforcement, the average road speed in New Zealand had decreased from 103 kmh to 96 kmh. That immediately got me thinking that maybe here is another cause for the increase in road toll with lower speed limits. The following explanation is a tad mathematical, but I’ll try and keep it in laymen’s’ terms with examples.
Imagine that there is a section of road network with intersections, bends, road works, new seal, some hills and rolling country, and passing through farmland, forestry, river gorges and coast line. In other words, typical New Zealand open roads. Irrespective of the number of vehicles in this road network travelling at 103 km, as one car enters the section, another will leave, so the number of vehicles remains constant. Let’s call this number n. If the average speed is reduced from 103 to 96 then the number of vehicles on the network at any given time will increase by 103/96 x n, or an increase of just over 7%.
Now take the situation where there is only 1 vehicle on the network, or n=1.
There is a small chance that this vehicle will crash. It may run off the road, hit a sheep that is on the road, the driver may get stung by a wasp, suffer sun strike, etc. There are many reasons why an accident may occur, but the chance is low. Let’s give this chance a probability Pr(a) (Probability of an accident) where Pr(a) is very small, maybe 0.0001 or 1:10,000 or less. Let’s also give a probability of colliding with another vehicle and call this Pr(c) (Probability of an collision) where Pr(c) is also very small, maybe 0.0001 or 1:10,000 or less. It doesn’t really matter just what values we use for this exercise, just that they are very small.
As there are no other vehicles on this road network, there is no chance of a collision with another, so the chance that this vehicle will have an accident is simply nPr(a) or 1 x 0.0001.
If there are 2 vehicles on this road network, n=2, then there is a chance that either (or both) could have a single vehicle accident. The chances of an accident have increased to 2 x Pr(a), but there is now a chance that they could hit each other or 1 x Pr(c). So now the chances of an accident have not doubled, but have trebled.
Lets add a third vehicle, n=3, and the chance of a single vehicle crash has now increased to 3Pr(a), but the chance of a multi vehicle accident is more complex. Collisions could occur between vehicles A and B, A and C, or B and C, (shown as AB, AC, BC). So the chances of a crash are now 3Pr(a) + 3Pr(c), or 6 times the chance as when only a single vehicle was on the road.
Lets add a fourth vehicle, n=4, and the chance of a single vehicle crash has now increased to 4 Pr(a), but the chance of a multi vehicle accident is increasing even faster. Collisions could occur between vehicles AB, AC, AD, BC, BD, and CD. So the chances of a crash are now 4 Pr(a) + 6 Pr(c), or 10 times the chance as when only a single vehicle was on the road.
Lets add a fifth vehicle, n=5, and the chance of a single vehicle crash has now increased to 5 Pr(a), but the chance of a multi vehicle accident is still increasing even faster. Collisions could occur between vehicles AB, AC, AD, AE, BC, BD, BE, CD, CE and DE. So the chances of a crash are now 5 Pr(a) + 10 Pr(c), or 15 times the chance as when only a single vehicle was on the road.
The chance of a single vehicle accident is increasing linearly with the number of vehicles on the road, but the chance of a collision with another vehicle is increasing at almost exponential rate. There is a Statistical measure for this rate called nCr , where nCr is the number of combinations that can be taken from n, r at a time. So for collisions between 2 vehicles, r would be 2, and the series would increase 0, 1, 3, 6, 10, 15, 21, 28, 36, 45 etc.
So, if there were 100 vehicles the multiplier would be 4950, and the chance of an crash is now 100 Pr(a) + 4950 Pr(c). Using Inspector Griffiths numbers of an average speed decrease from 103 to 96 then the number of vehicles on that road network would now be 107, and the chances of a crash have increased to 107Pr(a) + 5671Pr(c).
What this means is that decreasing the average speed by 7% has increased the chances of an accident by 14.4%. To keep the road toll down, the police should be trying to decrease the chances of crashes occurring, not increase them.
(The maths are actually a little bit trickier than this and I should be looking at the chance of not having an accident, but for such small values of Pr this gives a reasonable approximation of the true values)
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Originally Posted by Jantar
BTW, speed was a factor as both cars were moving, all be it at little more than walking pace.
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