Question

A terrible new virus has been discovered amongst​ beef-cattle in Southern Alberta. It is estimated that 5% of all​ beef-cattle are infected with this virus. A team of veterinarians have developed a simple test. Indications are that this test will show a positive result​ - indicating the​ beef-cow being tested has the virus​ - with a probability of 0.93 ​Unfortunately, this test has a​ false-positive probability of 0.08

​(a) A​ beef-cow in Southern Alberta was randomly chosen and given this test. The test results were​ positive, indicating the​ beef-cow has the virus. What is the probability that this particular​ beef-cow actually does have the​ virus?

​(b) What is the probability that a​ beef-cow that tests negative for this​ virus, actually has the​ virus?

Answer 1

a)

p(tested +) = p(virus and tested positive ) + p(not having virus and tested +)

= 0.05 * 0.93 + (1-0.05) * 0.08

= 0.0465 + 0.076

0.1225

so, p(beef-cow actually have the virus given tested positive)= p(virus and postive) / p(tested +)

= 0.05*0.93 /0.1225

0.379592

b)

P(tested negative)   =1-P(tested positive)

= 1-0.1225=0.8775

hence P(actually has the​ virus given tested negative

=P(virus and tested negative)/P(tested negative)

=0.05*(1-0.93)/0.8775

=0.0039886