It occurred to me today that a rearrangement of a classic numerical "trick"
usually used for mental calculation of interest rate returns could be put to
more honourable , or fun, purposes.

Simplistically, if an object is subject to a series of N trials with the
probability of destruction at each trial being x  (0 < x < 1) then the
probability of surviving the N trials is

              Survival = (1-X/100)^N

This is easily calculated with a calculator but not so easy to work in your
head for larger values of N.

However, a surprisingly good result for a good range of values of N and x is
given by


            Survival = 1 / 2 ^ (x.N / k )

k varies for best accuracy depending on the values of x and N.
A good figure for typical x and N values is k=70, so

             Survival = 1 / 2 ^ (x.N / 70)


As all competent embedded development engineers can calculate powers of 2
mentally, even for fractional indices, and as most other people liable to be
interested can multiply by 2 repeatedly, this becomes almost useful.

eg

G for George is a Lancaster bomber, now resident in the Canberra war
memorial museum in Australia.
G for George flew 90 combat missions over Europe. (Of the survivors only S
for Sally at 137 odd flew more).

If the average mission loss rate was 5% (about right on average, highly
variable in practice) what are the chances of G for George having survived?

90 missions x 5 percent / 70 ~= 6.4
2^6.4 ~= 80 (actually 84.4)
1/80 = 1.25%

Standard calculation gives (1-5/100)^90 == 1%.

Not too bad for a mental calculation.
Using the correct 84.4 gives 1.2%

Use with care and some [pretesting if of value to you. Errors can blow way
out for certain areas of input space. Adjusting k for your area of interest
helps.

The "method" uses the classic "rule of 72" - here modified to 'rule of 70'
:-).

If Scott Dattalo sees this he will no doubt provide a far superior method in
moments.

Working out how and why the rule of 72 works should give people many an hour
of mindless diversion,.




              Russell
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