Probability of having a disease and testing positive for it In a random test you test positive for a disease.In  $5$%  of  cases,  the  test  shows positive  even  when  the  subject does  not  have  the  disease. In  the  population  at  large,  one person  in  $1000$  has  the  disease. What  is  the  conditional  probability  that you  have  the  disease  given  that  you have  been  tested  positive,  assuming that  if  someone  has  the  disease,  he  will test  positive  with  probability  $1$? 
My approach was
$P($Teating positive)=P(test positive).$P($you dont have the disease)+$P($test positive).$P($you have the disease)$$=\frac {999}{1000}.\frac{5}{100}+\frac {1}{1000}=\frac {5095}{100000}$$
P(You have the disease$|$you test positive)=$\frac {P(\text{you have the disease,you test positive})}{P(\text{test positive})}$
$$=\frac {1}{1000}.\frac{100000}{5095}=\frac {20}{1019}$$
Is this correct?
Considering, 1000 people were taken for the test.  So, 50 people were positive by test and 1 person has the disease( as well as positive). So, the total test positives = 51 and true positive= 1. So the probability of having the disease and being positive = 1/51. Shouldn't this be the answer?
Considering, 1000 people were taken for the test.  So, 50 people were positive by test and 1 person has the disease( as well as positive). So, the total test positives = 51 and true positive= 1. So the probability of having the disease and being positive = 1/51. Shouldn't this be the answer?