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?

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