Answer to Question #119431 in Statistics and Probability for MLT

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 probability that the shipment came from Ghana

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

Ineffective vials

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

n=30

The probability of 1 ineffective vial

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

=0.2138

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

=0.3721

The probability of getting an ineffective vial

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

=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