The probability of getting a certain disease is 0.2. A certain test is available which returns a positive result 90% of the time for those with the disease, and a negative result 75% of the time for those without the disease. Compute Pr{Test positive | No disease}. (Hint: It may be helpful to draw a probability tree, and round to two decimal places)
To compute Pr{Test positive | No disease}, we need to consider the probability of getting a positive test result given that you do not have the disease.
Let's use a probability tree to break down the different possibilities.
First, let's define the following probabilities:
- Pr(Disease) = 0.2 (probability of having the disease)
- Pr(No disease) = 1 - Pr(Disease) = 0.8 (probability of not having the disease)
- Pr(Positive test | Disease) = 0.9 (probability of getting a positive test result given having the disease)
- Pr(Negative test | No disease) = 0.75 (probability of getting a negative test result given not having the disease)
Now, let's construct the probability tree:
0.2 (Disease)
/ \
0.9 (P) 0.1 (N)
/ \
0.8 (No disease) 0.2 (Disease)
/ \ / \
0.25 (N) 0.75 (P) 0.9 (P) 0.1 (N)
In the above probability tree, the top branch represents having the disease, and the bottom branch represents not having the disease. Each leaf node represents a different combination of having or not having the disease and getting a positive or negative test result.
We need to find Pr{Test positive | No disease}. From the probability tree, it is clear that the only relevant branch is the bottom left branch, which represents not having the disease and getting a positive test result. The probability associated with this branch is 0.75.
Therefore, Pr{Test positive | No disease} = 0.75.