On Sep 29, 2011, at 4:02 AM, Man on Bridges wrote:
On 29-9-2011 8:27, Horace Heffner wrote:
Looking at the other side of the coin, the probability of
catastrophic failure, suppose there is a 0.1% chance per hour one
of the E-cats can blow up spreading steam throughout the
container. There is thus a 0.999 probability of success, i.e. no
explosion for one E-cat, operating for one hour. The
probability that all 52 E-cats perform successfully for a 24 hour
test period is then 0.999^(52*24) = .287. That means there is a
71.3% chance of an explosion during a 24 hour test.
Me thinks you are wrong. Your statistical probability calculation
is based upon the fact that the chance of a single Ecat exploding
is influenced by it's behaviour earlier,
This is false. The probability in each time increment is assumed to
be independent. For there to be success there must be no failures for
any time increment. If there are T time increments, and the
probability of failure in any time increment is p, the probability of
success q=1-p in each time increment is independent of the other time
increments, and the probability of success in all time increments is
q^T (only possible if what happens in each time increment is
independent event), and the probability of any failure having
occurred is thus 1-(q^T).
which of course is not true. Statistically each Ecat has it's own
independent chance of explosion at any given moment which does not
change over time.
The instantaneous probability of failure is zero. Zero time results
in zero probability because lim t->0 q^t = 1 for for all 0=<q<=1 and
positive t. Therefore lim t->0 1-(q^t) = 0. Note that I provided an
assumption of 0.001 percent probability of failure *per hour*.
With your probability of 0,1% chance per hour this would result for
the whole of 52 Ecats then in a chance of explosion at any given
moment of 1 - (0.999^52) = .05 or 5%.
No. The probability of at least one E-cat failure in the 52 E-cat
system, based on the assumption of 0.001 probability of failure of an
individual E-cat in an hour is 1-(0.999)^52 = 0.506958 = 5%. Your
number 5% is right, but your interpretation of it representing an
instantaneous moment is wrong.
Looking even a bit more closer again this would mean that if the
chance of explosion is 0.1% per hour then the chance of explosion
is 2,77e-7 per second at any given moment for a single Ecat, which
would result for 52 Ecats into 1-((2,77e-7)^52) = 0,00001444434 or
0,00144% at any time.
The phrase "at any time" makes the above statement nonsensical.
An hour represents 3600 seconds, which are 3600 independent events of
1 second duration. Let a be the probability of failure in 1 second,
and b=(1-a) be the probability of success in 1 second. We have the
given probability p of failure for 3600 seconds being 0.001, and the
probability of success of one E-cat for one hour being q = 0.999.
The probability of success (no failures) for the 3600 1 second
independent time increments is
q = 0.999 = b^3600
b = q^(1/3600) = 0.999^(1/3600)
a = 1 - 0.999^(1/3600) = 2.779x10^-7
Note that a is the probability of failure in one second, not "at any
time". This is totally consistent with the probability of failure in
one E-cat in one hour being 5%. In other words, going backwards:
p = 1-(1-a)^3600 = 1-(1-2.779x10^-7)^3600 = 1-0.999 = 0.001
My calculations are therefore self consistent. The time intervals
are all treated as independent events. Your interpretation of
"moment" is perhaps a conceptual problem.