Question

Assume that a COVID-19 test can correctly diagnose 99% of non-infected people and 87% of infected people. Suppose that, in a population tested, 5% of people are infected. What is the probability that a person who tested positive is indeed infected?

Answer 1

D be an event of of being infected with the desease

So, P(D) = 0.05

be an event of of being non-infected with the desease.

So, P() = 1- 0.05 = 0.95

Let T be an event that the person diagnosed positive.

be an event that the person diagnosed negative

Test correctly diagnose 99% of non-infected people.

So, P( | ) =0.99

Test correctly diagnose 87% of infected people.

So, P( T | D ) =0.87

Probability that a person who tested positive is indeed infected

= P( D | T)

= P( T| D ) * P(D) / [ P( T| D ) * P(D) + P( T | ) * P( ) ] ( From Bayes' theorem)

=P( T | D) * P(D) / [ P( T| D ) * P(D) + {1- P( | )} * P( ) ]

= ( 0.87 * 0.05) / [ 0.87 * 0.05 + ( 1 - 0.99) * 0.95 ]

= 0.0435 / ( 0.0435 + 0.0095 )

=0.821

Probability that a person who tested positive is indeed infected = 0.0821

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