Question 590337: probability of failure of compnent is 0.06. Than what is the probability of failure of 109 components
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! if the probability of failure of one component is .06, then the probability of success of one component is 1 - .06 = .97
if you take a sample of n components, then the formula for the number of failures in that sample would be:
p(x failures) = C(n,x) * p^x * q^(n-x)
C(n,x) equals the combination formula of n! / (x! * (n-x)!)
p^x equals the probability of failure raised to the power of x.
q^(n-x) equals the probability of success raised to the power of (n-x).
example:
you draw a sample of 3 components.
p(0 failures) = C(3,0) * .06^0 * .94^3 which equals .830584
p(1 failure) = C(3,1) * .06^1 * .94^2 which equals .159048
p(2 failures) = C(3,2) * .06^2 * .94^1 which equals .010152
p(3 failures) = C(3,3) * .06^3 * .94^0 which equals .000216
total of all possible probabilities is equal to: 1.00000
the sum of all probabilities is equal to 1 as it should be.
if you draw a sample of 1000 components, then the sum of all probabilities will still be equal to 1 but each possible probability will be much less.
example:
the probability of exactly 106 defective components out of the sample of 1000 components is equal to:
C(1000,106) * .06^106 * .97^894 which is equal to ?????
my calculator is unable to calculate this.
it tells me that C(100,106) is too large for it to handle (overflow error).
it tells me that .06^106 is equal to 0.
it tells me that .94^894 is equal to 9.468934625 * 10^(-25)
suffice it to say that the probability of getting 106 defective components out of a batch of 1000 components is extremely small.
try something like 50 failures out of a batch of 100 components.
you get:
p(50 failures) = C(100,50) * .06^50 * .94^50 which is equal to 3.696656602 * 10^(-34)
that's a very small number.
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