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14 February, 13:02

Samples of laboratory glass are in small, lightpackaging or heavy, large packaging. Suppose that 2% and 1% of thesample shipped in small and large packages, respectively, breakduring transit. (a) If 60% of the samples are shipped in largepackages and 40% are shipped in small packages, what proportion ofsamples break during shipment? (b) Also, if a sample breaks duringshipment, what is the probability that it was shipped in a smallpackage?

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  1. 14 February, 13:20
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    a) 1.4% of the samples break during shipment

    b) the probability is 4/7 (57.14%)

    Step-by-step explanation:

    a) defining the event B = the sample of laboratory glass breaks, then the probability is:

    P (B) = probability that sample is shipped in small packaging * probability that the sample breaks given that was shipped in small packaging + probability that sample is shipped in large packaging * probability that the sample breaks given that was shipped in large packaging = 0.40 * 0.02 + 0.60*0.01 = 0.014

    b) we can use the theorem of Bayes for conditional probability. Then defining the event S = the sample is shipped in small packaging. Thus we have

    P (S/B) = P (S∩B) / P (B) = 0.40 * 0.02 / 0.014 = 4/7 (57.14%)

    where

    P (S∩B) = probability that sample is shipped in small packaging and it breaks

    P (S/B) = probability that sample was shipped in small packaging given that is broken
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