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 $G=$ the event that the shipment is from Ghana. Then $G^C$ is the event that the shipment is from neighbouring countries.

Given

$$P(G)=0.2$$

Then

$$P(G^C)=1-P(G)=1-0.2=0.8$$

Let $I=$ be the event that the vial is ineffective.

Use binomial distribution.

$$P(X=x)=\binom{n}{x}p^x(1-p)^{n-x}$$

Find the probability that one out of 30 vials is ineffective, given that the shipment is from Ghana

$n=30, p=0.1,x=1$

$$P(I|G)=\binom{30}{1}(0.1)^1(1-0.1)^{30-1}\approx0.14130386$$

Find the probability that one out of 30 vials is ineffective, given that the shipment is from neighbouring countries

$n=30, p=0.02,x=1$

$$P(I|G^C)=\binom{30}{1}(0.02)^1(1-0.02)^{30-1}\approx0.333970$$

We are looking for $P(G|I).$ Use Bayes' Theorem

$$P(G|I)={P(I|G)P(G)\over P(I|G)P(G)+P(I|G^C)P(G^C)}\approx$$

$$\approx{0.14130386(0.2)\over0.14130386(0.2)+0.333970(0.8)}\approx0.0957$$

$P(G|I)=0.0957$