Modelling False Positives

Suppose 1 in every 10,000 people are infected with a certain disease. The disease has an an associated test which is not completely accurate.
90% of infected people test positive.
10 of infection free people test positive.
We are interested in the probability that a person who test positive is infected. Of 10,000 people, 10 will be infected on average and 99,990 will be infection free..

We everyone is tested, and the Venn diagram is revised.

999 of the people who test positive are not infected.
The probability of actually being infected when you test positive is only  .