Disco Fever and Two Kinds of Tests

Question

25% of the people in a city have Disco Fever. There are two tests for Disco Fever, Rapid Antigen Test (RAT) and Full Scan Test (FST). When a person goes to a doctor to test for Disco Fever, with probability 2/3 the doctor prescribes RAT, and with probability 1/3 the doctor prescribes FST.

When RAT is done on a person, the outcome is as follows: If the person has the disease, the result is positive with probability 3/4. If the person does not have the disease, the result is positive with probability 1/4.

When FST is done on a person, the outcome is as follows: If the person has the disease, the result is positive with probability 1 (that is, FST has no false negatives). If the person does not have the disease, the result is positive with probability 1/3.

A person is picked uniformly at random from the town and is sent to a doctor to test for Disco Fever. The result comes out positive. What is the probability that the person has Disco Fever?

Answer 1

The question asks for P( disease | +RAT or +FST ).

The following probabilities are known:

In general:
- P( disease ) = 1/4

For the RAT:
- P( RAT used ) = 2/3
- P( +RAT | disease ) = 3/4
- P( -RAT | disease ) = 1/4
- P( +RAT | no disease ) = 1/4
- P( -RAT | no disease ) = 3/4

For the FST:
- P( FST used ) = 1/3
- P( +FST | disease ) = 1.0
- P( -FST | disease ) = 0.0
- P( +FST | no disease ) = 1/3
- P( -FST | no disease ) = 2/3

Several intermediate steps need to be calculated:

P( +RAT or +FST | disease ) = P( +RAT | RAT used, disease ) * P( RAT ) + P( +FST | FST used, disease ) * P( FST )
= 3/4*2/3 + 1*1/3
= 1/2 + 1/3
= 5/6

P( +RAT | RAT used ) = P( +RAT | RAT used, disease ) * P( disease ) + P( +RAT | RAT used, no disease ) * P( no disease )
= 3/4*1/4 + 1/4*3/4
= 3/16 + 3/16
= 3/8

P( +FST | FST used ) = P( +FST | FST used, disease ) * P( disease ) + P( +FST | FST used, no disease ) * P( no disease )
= 1.0 * 1/4 + 1/3*3/4
= 1/4 + 1/4
= 1/2

P( +RAT or +FST ) = P( +RAT | RAT used )*P( RAT ) + P( +FST | FST used )*P( FST )
= 3/8*2/3 + 1/2*1/3
= 3/12 + 1/6
= 5/12

Now, for the probability that the individual has Disco Fever:

P( disease | +RAT or +FST ) = ( P( +RAT or +FST | disease ) * P( disease ) ) / P( +RAT or +FST )
= ( 5/6 * 1/4 ) / ( 5/12 )
= 1/2

The individual has a 50% probability of having Disco Fever.

Comments

I would suggest to clean up the notation by using events like "positive RAT" "negative RAT" "positive FST" "negative FST".  As it is currently written, the random variable "positive test" is not defined unambiguously.

Answer 2

P(RAT)=2/3    P(FST)=1/3  P(disease)=1/4

RAT => P(+ve|disease) = 3/4    P(+ve|Nodisease) = 1/4

FST => P(+ve|disease) = 1    P(+ve|Nodisease) = 1/3

P(disease|+ve) = P(+ve|disease)*P(disease) / [P(+ve|diasease) + P(+ve|Nodisease)]

= [[P(RAT)*P(+ve|disease)+P(FST)*P(+ve|disease)]*P(disease)] / [[P(RAT)*P(+ve|disease) +P(FST)*P(+ve|disease)]*P(disease)+[P(RAT)*P(+ve|Nodisease)+P(FST)*P(+ve|Nodisease)]*P(Nodisease)]

= [[2/3*3/4+ 1/3*1]*1/4] / [[2/3*3/4 + 1/3*1]*1/4 + [2/3*1/4 + 1/3*1/3]*3/4]

= [1/8 + 1/12] / [1/8 + 1/12 + 1/8 + 1/12]

= [1/8 + 1/12] / 2*[1/8 + 1/12]

=1/2

Therefore, probability the person has disco fever given the result is positive is 0.5