Probability Situation > A well known medical research organization conducted a study on a certain brand of test strips to see if a person has a certain disease. They found that $32\%$ of the people using the test strips actually have the disease. When a person with the disease uses these strips it correctly indicates that they have the disease $97\%$ of the time. A person who does not have the disease gets a correct reading $88\%$ of the time. Find $P(\text{has disease}\ \mid \text{tests positive for it})$ My attempt is as follows. Using Bayes' Rule: $$P(\text{has disease}\ \mid \text{tests positive})=\frac{P(\text{has disease } \cap\text{test positive})}{P(\text{tests positive})}$$ We know the value of the numerator to be $0.97$ from the info in the passage. But how do we find the value of the denominator from the information given.

Suppose $10,000$ people are tested (to avoid decimals)

Of these, $32\%$ or $3200$ have the disease, of which $97\%$ or $3104$ test positive

$6800$ don't have the disease, of which $12\%$ or $816$ test positive

Thus P(have the disease | test positive) $= \dfrac{3104}{3104+816}$