# Gas station without pumps

## 2010 November 21

### What are the odds of killing 2 bicyclists?

Filed under: Uncategorized — gasstationwithoutpumps @ 00:05
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A truck driver was recently involved in a crash which killed a bicyclist for which there were no witnesses.  It turns out that he had been involved in a crash a few years ago that killed a bicyclist.  What are the odds that he is just unlucky (as opposed to the alternative hypothesis that he is a bad truck driver)?

Here is a news story on the second crash:

Truck driver in Alpine Road collision also involved in fatal 2007 crash in Santa Cruz
By Jesse Dungan

The driver of a big rig that killed a female bicyclist near Portola Valley earlier this month was involved in a similar crash in Santa Cruz three years ago that left a popular high school teacher dead, a California Highway Patrol spokesman confirmed Monday.

The driver, Gabriel Manzur Vera, was determined not to be at fault in the Santa Cruz collision, said CHP Officer Art Montiel. Vera also has not been charged in the Nov. 4 crash on Alpine Road, he said. …

The truck driver’s employer recently settled a wrongful-death suit for the first death, paying \$1.5 million.

How can we estimate the odds and determine whether this can reasonably be expected as a chance event?

As always when we want to compute the probability of a chance event, we need to define our null model carefully, and make sure that we use appropriate corrections for multiple hypotheses.  We probably want our null models to slightly overestimate the probability of killing bicyclists, so that our models will err on the side of saying that the event of one driver killing 2 bicyclists is not strange enough to be statistically significant.

Let’s look at two simple null models, one based on number of drivers, one based on miles driven.

First, how likely is any driver to kill a bicyclist in a given year? In their lifetime? What about the probability of killing two bicyclists in separate incidents in their lifetime?  How many such double-death motorists should we expect to find in the US?

Second, how likely is a driver to kill a bicyclist for each mile driven? How many miles does a truck driver drive in a year?  How likely does that make it for a given truck driver to kill a bicyclist in a year?

If the probability that a driver kills a cyclist in a year is $p$, and the events are independent events, unrelated to the driver’s skill, then the chance that a driver will ever kill a cyclist in a 60-year driving career is $1-(1-p)^{60}$ and the chance that a driver will kill 2 or more cyclists in separate years is about $1 - (1-p)^{60} - 60 p (1-p)^{59}$, which we can rewrite as $1- (1-p)^{59}(1+59p)$. I picked a 60-year career as an over-estimate for most truck drivers, as we are trying to overestimate the number of deaths.

There were about 716 bicyclists killed in motor vehicle crashes in 2008 [http://www.bts.gov/publications/national_transportation_statistics/html/table_02_19.html] The number fluctuates from about 630 to 830 over the past two decades, so 716 seems fairly typical.  But since we want to overestimate the probability of killing bicyclists, let’s round up to 990 a year, bigger than BTS has recorded since about 1975.

In 2006 (the latest year I could find numbers for) there were 202,810,438 licensed drivers [http://www.fhwa.dot.gov/policyinformation/pubs/pl08021/excel/Fig4_1.xls].  The number was probably higher in 2008, but we are trying to overestimate the probability of killing a bicyclist, so we should underestimate the number of  drivers.  With this number and the overestimate of bicyclist deaths, we can estimate that one bicyclist is killed each year per 205,000 drivers. The chance that a driver ever kills a cyclist is about 0.0003.  (Double-checking, that comes to 60,843 cyclists killed in 60 years, a high number, but consistent with our over-estimates.)  The chance that a driver kills two cyclists in separate incidents over a 60-year career is about 1 in 23,700,000, so the expected number of people who would do that is about 8.5.  Using this model, we would expect about 8 or 9 double-killers in the US in 60 years, so the event is rare, but not so rare as to immediately claim that the event of killing 2 cyclists is proof of incompetent driving. (It certainly raises that suspicion very strongly, though).

There were about 2,973,509 million miles driven in 2008 [http://www.bts.gov/publications/national_transportation_statistics/html/table_01_32.html], so there is a bicyclist killed for about every 3 billion miles driven.

We’re trying to overestimate the probability of a truck driver killing a bicyclist, so we need to overestimate (somewhat) the number of miles a truck driver drives.  Let’s assume he drives 10 hours a day, 364 days a year at 55 mph.  That would be about 200,000 miles and over 60 years 12 million miles would be driven (this is a somewhat large overestimate). There are about 3.5 million truck drivers driving 400 billion miles a year [http://www.truckline.com/About/Industry/Pages/ProfessionalTruckDrivers.aspx], or about 114,000 miles a year, so the lifetime driving of a truck driver is only over estimated by a factor of 2–4 times.

The probability of a particular truck driver ever killing a bicyclist is about $1- (1-1/3000000000)^{12000000}$ (treating each mile driven as independent), which we can estimate by using the approximation $(1-x/N)^N \approx e^{-x}$ for large $N$.  So the probability of a truck driver ever killing a bicyclist is about 1 in 250.  Checking, we would expect 3.5 million/250 or 14000 bicyclists killed by trucks over 60 years in the US.  I don’t have numbers for bikes killed by trucks, but 233 a year seems reasonable, given the number killed by all motor vehicles combined.

The probability of killing 2 or more bicyclists in separate miles is $1 - (1-p)^{N} - N p (1-p)^{N-1}$, where $p$ is the probability of killing in any given mile, and \$N\$ is the number of miles. We can rewrite that as $1 - (1-p)^N (1+Np/(1-p))$, which for our truck-driver estimates is about  1 in 125,000.  With 3.5 million truck drivers around, we would expect to find about 28 double-bicyclist-death truck drivers by chance in 60 years.

With both models, using overestimates of how likely bicyclists are to be killed by chance, we get an expected number of chance double-death truck drivers of between 1 such driver every 2 years in the country and one every 8 years.  This means that we can’t completely reject the null model (that the driver was just unlucky enough to have 2 chance encounters), but our suspicions about the driver should certainly be raised, and other evidence checked to see whether the driver is really incompetent to be driving a truck.

Late-breaking news: It seems that Gabriel Vera was involved in 3 fatal accidents, not just two, so there is no question in my mind that he isn’t just an unlucky driver, but should never be allowed to drive again.

1. This was a fantastic post, and I thank you for sharing it.
I am very disturbed by both accidents, but I am also concerned with the investigations of the collisions and possible bias. The reporting of the Nov. 4 fatality has also been deplorable.
Anyone know what exactly is available to the public with regards to information about the investigations? Too many things aren’t adding up, but its possible that is due to the lack of comprehensive information being reported.

Comment by Menlo Park Resident — 2010 November 23 @ 11:40

• Just an update from our local Almanac http://www.almanacnews.com/news/show_story.php?id=9662 for anyone following this story:

Investigation leads CHP to change findings on Alpine Road fatal bicycle-truck accident
Human DNA found near the front axle of a tractor-trailer truck has led investigators from the California Highway Patrol to conclude that the November 4, 2010, collision and death of Los Altos Hills cyclist Lauren Ward was not the fault of Ms. Ward, authorities said.

The accident re-enactment and subsequent investigation, which has been ongoing since December and is now over, does not conclude that the driver of the truck is at fault, Mr. Montiel told the Almanac.

Comment by Rebecca — 2011 September 21 @ 00:47

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