Each of 500 soldiers in an army company independently has a certain disease with probability 1/103. this disease will show up in a blood test, and to facilitate matters, blood samples from all 500 soldiers are pooled and tested. (a) what is the (approximate) probability that the blood test will be positive (that is, at least one person has the disease)

The chance of the individual has the disease is 1/103, which means there is 102/103 chance to be healthy. 
The chance of "at least one person has the disease" is the opposite of the chance of  "everyone healthy".

The chance that all 500 soldiers healthy would be: (102/ 103)^500= 0.00761155742

The chance of at least one person has the disease would be: 1- (102^500/ 103^500)= 0.99238844257 = 99.24%