## Thursday, March 18, 2010

### Probability of crashing a courtesy car

I have booked my Toyota Auris in for its 30,000 mile service on April 12th. And to have that irritating go-faster feature removed of course.

The garage offered me a courtesy car so I could continue on to work but there's a catch. If I have an accident I have to pay £1,000 excess. But on payment of £10 for an insurance deal the excess would be reduced to only £100.

What can be deduced from this about the probability - in the garage's opinion - that I will have a crash?

Let the crash probability be p. We assume the garage is indifferent as to which option I take (this would be their rational position) so I would end up paying the same to cover repair costs in either case. This is then a problem in expected values. 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 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?

I'll have to think about it!