## Your shipment of FAIL has arrived

I was reading “Steve Waddington’s Network Notes” today where he describes how two electrical components failed on him at the same time, and wonders what the chances were of it happening.

It is highly unlikely that two electrically isolated components would fail at exactly the same time. Since both are rated at an MTBF of 90,000 hours, the chance of them both failing in any given hour, after less than 10,000 hours of operation, would have to be in the region of one in one billion.

It got me wondering… what were the chances?

“MTBF” is “Mean Time Between Failures”.  It is the reciprocal of the failure rate, λ, and follows an exponential failure distribution.  This distribution is asymmetrical, so it is not true to say that the MTBF represents the point at which the probability of failure equals 50%.  However, an exponential distribution does make probability calculations relatively easy.

P(component fails at exactly 10000 hrs | MTBF=90000 hrs) = λe−λx

= 1/90000 * exp (-1/9)

= 0.00000994

or about one chance in 100,577.

That’s the chance of one component failing.  The chance of the two components failing at once is therefore one in 100,5772 or more than one in 10 billion.  No wonder Steve ruled out “just bad luck”.  I’d be very suspicious too.  It will be interesting to see if his theory of a dodgy part proves to be correct.