Answer to Question #119387 in Statistics and Probability for Michael

Question

A hospital receives 30% of its COVID-19 vaccine shipments from Ghana Health Service
and the remainder of its shipments from neighboring 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 neighboring 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 neighboring West
African countries?

Answer 1

We remind the Bayes' Theorem:

$P(A\,\,|\,\,B)=\frac{P(B\,\,|\,\,A)P(A)}{P(B)}$ ,

where A denotes the event that the shipment came from the neighboring West African countries

B denotes the event that from 30 randomly selected vials from a shipment one is ineffective.

$P(A\,\,|\,\,B)$ denotes the probability of event A in case B has happened.

$P(B\,\,|\,\,A)$ denotes the probability of event B in case A has happened.

From formulation of the problem we receive that $P(A)=0.7$,

For calculation of $P(B)$ we consider two cases:

(i) Assume that the shipment came from neighboring West African countries.

The probability that from 30 vials one is ineffective is:

$P(B\,\,|\,\,A)=C_{30}^1\cdot0.03\cdot0.97^{29}=30\cdot0.03\cdot0.97^{29}\approx0.3721$ ,

where $C_{30}^1$ denotes the binomial coefficient.

(ii) Assume that the shipment came from GHS.

The probability that from 30 vials one is ineffective is:

$C_{30}^1\cdot0.08\cdot0.92^{29}\approx0.2138$

Using the Law of total probability we get:

$P(B)=0.3\cdot0.2138+0.7\cdot0.3721=0.3246$

Then, we get

$P(A\,\,|\,\,B)=\frac{0.3721\,\cdot\,0.7}{0.3246}=0.8024$

Answer: 80.24%