On 11-10-03 12:38 PM, Man on Bridges wrote:
Hi,
On 2-10-2011 14:25, Horace Heffner wrote:
In other words, if there is any possibility of failure, i.e. q<1,
then repeated events eventually, much more quickly than ordinary
common sense dictates, result in failure. For example, if the
probability of a catastrophe when drinking and driving once is 1/1000
= .001, then the probability of *no* catastrophe in 100 such events
is 1-.999^100 = 0.095, or about 1%.
This should of course be read as:
then the probability of *a* catastrophe in 100 such events is
1-.999^100 = 0.095, or about 1%.
Both wrong. Where'd you guys get those 1% values, anyway?
1 - 0.001 = 0.999, sure enough.
But 0.999 ^ 100 = 0.904792147...., which means there's about 90% chance
of no catastrophe, or about 10% chance of a mess. Strangely, neither of
you got the 10% number, even though it's not just correct, it's also
what you'd guess if you didn't know anything (1 chance in 1000, repeated
100 times, give about 1 chance in 10 of hitting).
For raising probabilities near 1 to some power, to get an idea of what
the answer should be, one can use the truncated Taylor series. The
derivative with respect to the distance from 1 is:
d/dx ((1 - x) ^ k) = -k * (1 - x) = -k + k*x --> tends to -k as x
approaches 0
So for small values of x, discarding all but the linear term and using
the limiting value for the first derivative, we have
(1 - x)^k ~ 1 - k*x
which is, in fact, just about what you'd expect naively. In this
particular case, that yields:
1 - 0.001 * 100 = 0.9
which is in pretty good agreement with the exact value I gave above (but
lousy agreement with the values previously given -- neither of you guys
seems to have been paying much attention to the decimal point).
In general, it's only after things dip well below 90% that the "naive
formula" starts going seriously wrong.
