Answer to Question #119649 in Statistics and Probability for desmond

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

The probabilities that the shipments came from Ghana and neighboring West Africa countries are

$P(G) =0.3, P(G^c) =0.7$

respectively.

The probabilities of an ineffective vial coming from Ghana and neighboring countries are

$P(I|G) =0.08, P(I|G^c) =0.03$

Given n =30, the probability of getting one ineffective vial is

$P(I=1|G)={30\choose 1}0.08^1 0.92^{29} =0.2138$

$P(I=1|G^c)={30\choose 1}0.03^1 0.97^{29}=0.3721$

The probability of getting one ineffective vial is

$P(I=1) =P(I=1|G) P(G) +P(I=1|G^c) P(G^c)$

=0.2138*0.3+0.3721*0.7=0.3246

The probability that the shipment came from neighbouring West African countries is calculated by applying the Bayes theorem

$P(G^c|I=1)=\frac {P(I=1|G^c)P(G^c)}{P(I=1)} =\frac {0.3721 *0.7} {0.3246} =0.8024$