`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%."`

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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 ofcatastrophic failure, suppose there is a 0.1% chance per hour oneof the E-cats can blow up spreading steam throughout thecontainer. There is thus a 0.999 probability of success, i.e. noexplosion for one E-cat, operating for one hour. Theprobability that all 52 E-cats perform successfully for a 24 hourtest period is then 0.999^(52*24) = .287. That means there is a71.3% chance of an explosion during a 24 hour test.Me thinks you are wrong. Your statistical probability calculationis based upon the fact that the chance of a single Ecat explodingis influenced by it's behaviour earlier,This is false. The probability in each time increment is assumedto be independent. For there to be success there must be nofailures for any time increment. If there are T time increments,and the probability of failure in any time increment is p, theprobability of success q=1-p in each time increment is independentof the other time increments, and the probability of success in alltime increments is q^T (only possible if what happens in each timeincrement is independent event), and the probability of any failurehaving occurred is thus 1-(q^T).which of course is not true. Statistically each Ecat has it's ownindependent chance of explosion at any given moment which does notchange over time.The instantaneous probability of failure is zero. Zero time resultsin zero probability because lim t->0 q^t = 1 for for all 0=<q<=1and positive t. Therefore lim t->0 1-(q^t) = 0. Note that Iprovided an assumption of 0.001 percent probability of failure *perhour*.With your probability of 0,1% chance per hour this would resultfor the whole of 52 Ecats then in a chance of explosion at anygiven moment of 1 - (0.999^52) = .05 or 5%.No. The probability of at least one E-cat failure in the 52 E-catsystem, based on the assumption of 0.001 probability of failure ofan individual E-cat in an hour is 1-(0.999)^52 = 0.506958 = 5%.Your number 5% is right, but your interpretation of it representingan instantaneous moment is wrong.Looking even a bit more closer again this would mean that if thechance of explosion is 0.1% per hour then the chance of explosionis 2,77e-7 per second at any given moment for a single Ecat, whichwould result for 52 Ecats into 1-((2,77e-7)^52) = 0,00001444434or 0,00144% at any time.The phrase "at any time" makes the above statement nonsensical.An hour represents 3600 seconds, which are 3600 independent eventsof 1 second duration. Let a be the probability of failure in 1second, and b=(1-a) be the probability of success in 1 second. Wehave the given probability p of failure for 3600 seconds being0.001, and the probability of success of one E-cat for one hourbeing q = 0.999. The probability of success (no failures) for the3600 1 second independent time increments isq = 0.999 = b^3600 b = q^(1/3600) = 0.999^(1/3600) a = 1 - 0.999^(1/3600) = 2.779x10^-7Note that a is the probability of failure in one second, not "atany time". This is totally consistent with the probability offailure in one E-cat in one hour being 5%. In other words, goingbackwards:p = 1-(1-a)^3600 = 1-(1-2.779x10^-7)^3600 = 1-0.999 = 0.001My calculations are therefore self consistent. The time intervalsare all treated as independent events. Your interpretation of"moment" is perhaps a conceptual problem.Kind regards, MoBBest regards, Horace Heffner http://www.mtaonline.net/~hheffner/

Best regards, Horace Heffner http://www.mtaonline.net/~hheffner/