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/




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