The sentence below: "This is totally consistent with the probability
of failure in one E-cat in one hour being 5%." should read: "This is
totally consistent with the probability of failure of at least one E-
cat (of 52) in one hour being 5%."
On Sep 29, 2011, at 11:34 AM, Horace Heffner wrote:
On Sep 29, 2011, at 4:02 AM, Man on Bridges wrote:
Hi,
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.
Kind regards,
MoB
Best regards,
Horace Heffner
http://www.mtaonline.net/~hheffner/
Best regards,
Horace Heffner
http://www.mtaonline.net/~hheffner/
