Question 271761
1% is the same as .01


99% is the same as .99


when the process malfunctions 50 out of 100 items are defective.


this means that the probability that each item is defective is .5


this also means that the probability that each item is not defective is 1 minus .5 = .5


if you want to find the probability that at least 1 of the items is defective, then you need to find 1 minus the probability that 0 of the items are defective.


since the probability that each item is not defective is .5, we can work with that to determine how many items need to be tested to make the probability that 0 of the items are not defective be less than or equal to .01.


if the probability that 0 of the items are not defective is less than or equal to .01, that means that the probability that at least one of the items is defective would be greater than or equal to .99 which is what we are looking for.


if we take one sample, the probability that 0 of the samples is defective is .5
= .5^1


if we take two samples, the probability that 0 of the samples is defective is .5 * .5 = .5^2


if we take three samples, the probability that 0 of the samples is defective is .5 * .5 * .5 = .5^3


the formula we are looking for is:


.5^n <= .01 where n is the number of samples.


we take the log of both sides of this equation to get:


log(.5^n) <= log(.01)


this becomes:


n * log(.5) <= log(.01)


divide both sides of this equation by log(.5) to get:


n <= log(.01) / log(.5)


use your calculator to solve for n to get:


n <= 6.64385619


since n has to be an integer, then n has to be at least 7 for the probability that 0 of the items are defective to be <= .01


.5^6 = .015625
.5^7 = .0078125


the probability that at least 1 of the items is defective is equal to 1 minus the probability that 0 of the items is defective.


when 7 samples are taken, the probability that at least 1 of the samples is defective would be 1 - .0078125 = .9921875 which is greater than .99


when 6 samples are taken, the probability that at least 1 of the samples is defective would be 1 - .015625 = .984375 which is less than .99.


at least 7 items should be tested to have greater than or equal to 99% chance of at least 1 of them being defective.








the probability that for each sample 1 of the items is defective would be 50/100 = .5


this means the probability that for each sample 0 of the items is defective would be


if you draw a sample of 2, then the probability that 0 are defective is .5 * .5 = .5^2 = .25


sample of 3, then .5^3 = .125


sample of 4, then .5^4 = .0625


sample of 5, then .5^5 = .03125


you can make a general equation out of it.


the equation would be:


.5^x <= .01


take the log of both sides of this equation to get:


log(.5^x) <= log(.01)


because log(x^n) = n*log(x), this becomes:


x * log(.5) <= log(.01)


divide both sides by log(.5) to get:


x <= log(.01)/log(.5)


use your calculator to solve for x to get:


x <= 6.64385619


since samples have to be in whole numbers, then our answer should be that the number of samples has to be at least 7.


to test this out, we take .5^6 and .5^7 to see what we get.


.5^6 = .015625 which is >= .01


.5^7 = .0078125 which is <= .01


we need at least 7 samples to have at least a 99% probability that at least one of the samples will have at least 1 defective item.