Consider, for example, a population of 10,000 people and test everybody for the disease. The 95% accuracy of the test means that 95% of those with the disease test positive and 95% of those without it test negative. Thus, in the population, of the 100 who have the disease, 95 test positive and of the 9,900 who do not have it, 9,405 test negative. However, this means that 495 healthy people test positive in error, and among the total of 95+495=590 who test positive, only 95 (about 16%) actually have the disease.
To obtain a formal solution, you need to apply the celebrated Bayes' rule, which is also useful in court cases, for email spam filters, and for internet search engines.
The Reverend Thomas Bayes lived in England between 1702 and 1761. He never published his famous rule or any other mathematics. A brief biography is supplied by the International Society for Bayesian Analysis. I have made quite a surprising discovery regarding Bayes' portrait.