Question

A hospital receives 20% of its COVID-19 vaccine shipments from Japan and the
remainder of its shipments from neighbouring countries. Each shipment contains a very large number of vaccine vials. For Japan’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 Japan ?

Answer 1

H₁={from Japan}; p(H₁)=0.2

H₂={from neighbouring countries}; p(H₂)=0.8

A={1 is ineffective from 30 randomly selected}

$p(H_1/A)=\frac{p(A/H_1)p(H_1)}{p(A/H_1)p(H_1)+p(A/H_2)p(H_2)}$

$p(A/H_1)=C_{30}^1(0.1)(0.9)^{29}=0.1413$

$p(A/H_2)=C_{30}^1(0.02)(0.98)^{29}=0.334$

$p(H_1/A)=\frac{0.1413*0.2}{0.1413*0.2+0.8*0.334}=0.096$

Answer: 0.096.