Answer to Question #116047 in Statistics and Probability for Ulrich

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

Probability that the shipment is from Ghana is P(G) =0.2 hence $P(G^c)$ =0.8

P(I|G) =0.1, P($I|G^c)$ =0.02

The probability that the one ineffective out of the 30 shipments comes from Ghana follows a binomial distribution

$P(I=1|G)={30 \choose 1}0.1^1 *0.9^{29}$

=0.1413

$P(I=1|G^c)={30 \choose 1}0.02^1 *0.98^{29}$

=0.334

The probability that one shipment is ineffective is;

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

=0.1413*0.2+0.334*0.8

=0.29546

Using Bayes theorem, the probability that the ineffective shipment comes from Ghana is;

$P(G|I=1)=\frac{P(I=1|G)*P(G)}{P(I=1)}$

$=\frac{0.1413*0.2}{0.29546}$

=0.0956