Crashing The Courtesy Car

I stood forlornly on the freezing forecourt, buffeted by icy gusts. Looking up at the leaden pre-dawn sky with its scudding dark clouds, I reflected that it did indeed look very much like a TV screen tuned to a dead channel. I pulled my coat tighter against the flurries of snow and walked into the service center.

I had booked my car in for its 30,000 mile service a few weeks ago. After completing the paperwork and handing over the keys, the guy behind the desk got around to the Courtesy Car, which I would need to continue on to work. But there was a catch.

‘If you happen to have an accident,’ he said, ‘There’ll be a $1,000 excess to pay. But if you hand over $10 for an insurance deal right now, that excess will be reduced to only $100.’

I was thinking furiously: so they want me to pay them $10 but is it really value for money? How likely is it, I thought to myself, that I’m going to crash their car today? But perhaps there’s a different, more informative question: what does this guy’s proposal tell me about the garage’s assessment of my chances of damaging their vehicle?

Let the crash probability be p. I have to start with the assumption that the garage has set the pricing on this offer so that they’re indifferent to whatever option I end up taking. This would be their rational position, so I’d end up paying the same expected amount to cover repair costs whether I took the offer or not.

Let’s equalize my expected outlay on both sides of the deal – this is the amount I spend times the probability of the accident happening. So:

1000p = 10 + 100p

recalling that I have to pay them $10 regardless in the second case.
So clearly p = 1/90.

This seems a rather high probability of crashing. Driving 270 days a year at this probability I would expect to have three crashes a year (I don’t). But of course, I don’t drive an unfamiliar courtesy car every one of those 270 days. I believe the chances of my crashing their courtesy car are considerably less than 1.1% but would I pay $10 to avoid a very remote risk of paying an additional $900?

They obviously expect a continuing cash-flow from risk-averse drivers with this deal!

With the car rocking under the impact of successive wintry squalls, I drove into the morning traffic without handing over any cash; amazingly, I didn’t crash that day. I was, however, considerably confused by the way the engine turned itself off every time I halted at traffic lights: something about combating global warming, I believe.

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This would have been my last "science feature" article for sciencefiction.com, had they chosen to use it. They evidently thought it was not 'SFnal enough' and I have to agree.