Probability of correct detection given conditional accuracies

Question

Suppose $1\%$ of people suffer from a certain disease. The probability that this disease will be detected during a routine check-up is $0.9$, while the probability that a healthy person will be incorrectly diagnosed of suffering from the disease is $0.01\%$. If someone goes for a routine check up, and it is found that he has the disease, what is the probability that the diagnosis was wrong?

Answer 1

HINT: Let $P$ be the probability that you are trying to find. Then $$P=\frac{W}{D}$$ Where $W$ is the probability that the patient was free of the disease but was diagnosed as having it, and $D$ is the probability that the patient is diagnosed as having the disease. This can be expanded further as $$P=\frac{W}{W+C}$$ Where $C$ is the probability that the patient actually has the disease and is diagnosed as having it.

Answer 2

To start you on your way, express what you are given in symbols:

• Let $D$ represent the event for having the disease.
• Let $T$ represent the event for a positive test result.

Suppose $1\%$ of people suffer from a certain disease.

$\mathsf P(D)=0.01$

The probability that this disease will be detected during a routine check-up is $0.9$,

$\mathsf P(T\mid D) = 0.9$

while the probability that a healthy person will be incorrectly diagnosed of suffering from the disease is $0.01\%$.

$\mathsf P(T\mid D^\complement)=0.0001$

If someone goes for a routine check up, and it is found that he has the disease, what is the probability that the diagnosis was wrong?

Find $\mathsf P(D^\complement\mid T)$ the conditional probability that they don't have the disease given that the test says they do.