A hospital receives 30% of its COVID-19 vaccine shipments from Ghana Health Service and the remainder of its shipments from neighbouring West African countries. Each shipment contains a very large number of vaccine vials. For GHS’s shipments, 8% of the vials are ineffective and for the neighbouring countries, 3% of the vials are ineffective. The hospital tests 30 randomly selected vials from a shipment and finds that one is ineffective. What is the probability that the shipment came from neighbouring West African countries?
W - shipment from West Africa
I - one vial is ineffective
G - shipment from Ghana
P(W)=1-0.3=0.7
P(I|W)= "\\begin{pmatrix}\n\n 30 \\\\\n\n 1\\end{pmatrix}" *0.03*0.9729=0.372
P(I|G)="\\begin{pmatrix}\n\n 30 \\\\\n\n 1\\end{pmatrix}" *0.08*0.9229=0.2138
P=(P(I|W)*P(W))/(P(I|W)*P(W)+P(I|G)*P(G))=(0.372*0.7)/(0.372*0.7+0.2138*0.3)=0.8
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