Answer to Question #115123 in Statistics and Probability for Majoy

Question

A hospital receives 20% of its COVID-19 vaccine shipments from Ghana and the remainder of its shipments from neighbouring countries. Each shipment contains a very large number of vaccine vials. For Ghana’s shipments, 10% of the vials are ineffective. For the neighbouring countries, 2% 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 Ghana.

Answer 1

Let G be the event that shipment is from Ghana,

I - the event that the vial tested is ineffective.

$P(G)=0.2.$

$P(I|G)=C_{30}^10.1^10.9^{29}=0.1413.$

$P(I|notG)=C_{30}^10.02^10.98^{29}=0.3340.$

$P(G|I)=\frac{P(I|G)P(G)}{P(I|G)P(G)+P(InotG)P(notG)}=\frac{0.1413*0.2}{0.1413*0.2+0.3340*0.8}=0.0956.$