What is the probability that none of the 3 vials have hairline cracks?
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In a shipment of 87 vials, only 17 do not have hairline cracks. If you randomly select 3 vials from the shipment, what is the probability that none of the 3 vials have hairline cracks?
The probability of pulling three uncracked vials is 17/87 x 16/86 x 15/85 = .006415
You can use the hypergeometric distribution to find the solution
Let X be the number of vials with a hairline crack. X has the hypergeometric distribution with the following parameters.
K = number of items to be drawn = 3
N = total objects = 87
M = number of objects of a given type = 17
The probability mass function for the hypergeometric distribution is defined as:
P(X = x | N, M, K) = ( M C x ) * ( (N – M) C (K – x) ) / ( N C K )
for x = {0, …, K}; M – (N – K) ≤ x ≤ K
P(X = 0 | N, M, K) = 0 otherwise
Note that the constraints on x here are very generic and it is possible to have value of K, N and M such that for x in {0, …, K} P(X = x) = 0.
If you have n objects and chose r of them, the number of combinations is:
n! / ( r! (n-r)! )
this can be written as nCr
the N C K is the total number of possible combinations of K objects drawn from N objects.
the M C x is the number of combinations of getting x objects of the given type
the (N – M) C ( K – x) is the number of combinations of non typed objects to be drawn.
Looking at the PMF you should be able to see that it is the ratio of the number of combination of selecting the X of the items of interest times the number of combinations of choosing K – X items from the remaining items and this is all divided by the total number of combination for choosing K items from N objects.
The expectations of the Hypergeometric distribution is KM / N = 0.5862069
The variance is K (M/N)(1-M/N)(N-K)/(N-1) = 0.4606919
The std dev is the sqrt of the variance: 0.6787429
The Probability Mass Function, PMF,
f(X) = P(X = x) is:
P(X = 0 ) = 0.5164395 <<<< ANSWER
P(X = 1 ) = 0.3873296
P(X = 2 ) = 0.08981556
P(X = 3 ) = 0.006415397
It depends on the cause of the crack – if they were caused by an impact, the vials with cracks are likely to be closer together. If the cracks are due to manufacture, each is equally likely to be faulty, as the likelihood would be independent of where each vial is placed.