Answer to Question #119817 in Statistics and Probability for Asubonteng

Question

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?

Answer 1

W - shipment from West Africa

I - one vial is ineffective

G - shipment from Ghana

P(W)=1-0.3=0.7

P(I|W)= $\begin{pmatrix} 30 \\ 1\end{pmatrix}$ *0.03*0.97²⁹=0.372

P(I|G)=$\begin{pmatrix} 30 \\ 1\end{pmatrix}$ *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