Agenetic test is used to determine if people have a predisposition for thrombosis, which is the formation of a blood clot inside a blood vessel that obstructs the flow of blood through the circulatory system. it is believed that 3% of people actually have this predisposition. the genetic test is 99% accurate if a person actually has the predisposition, meaning that the probability of a positive test result when a person actually has the predisposition is 0.99. the test is 98% accurate if a person does not have the predisposition. what is the probability that a randomly selected person who tests positive for the predisposition by the test actually has the predisposition?

The probability that a randomly selected person that tests positive for the predisposition by the test actually has the predisposition is : 0.6

i.e. P( Disease | positive ) = 0.6

Let : B = positive , A = disease

∴ P( A | B ) = P( B | A ) P ( A ) / P( B )

where : P( B | A ) = 0.99

             P ( A ) = 0.03

∴ P(Disease | positive ) = P ( positive |  disease ) * P( disease ) / P ( positive )

                                        =   (0.99) * (0.03) / P(positive )   ( 1 )  

Also  P( Positive ) = P( B | A ) P( A )+ P( B | no disease

)* P( no disease )  

                             = (0.99)*(0.03) + (0.02)* (0.97) = 0.049 ( 2 )

back to equation ( 1 )

P( Disease | positive ) = (0.99)* (0.03) / (0.049) = 0.6.

Hence the probability that a randomly selected person person who tests positive has the predisposition is 0.6

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