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)
0.2
0.18
To compute Pr{Test positive | No disease}, we need to use conditional probability.
Let's break down the problem step by step:
Step 1: Define the events
Let A be the event of having the disease. The probability of having the disease is given as Pr(A) = 0.2 (or 20%).
Let B be the event of testing positive for the disease. We are interested in finding Pr(B | A'), which represents the probability of testing positive given that there is no disease. A' denotes the complement of A.
Step 2: Calculate Pr(B | A)
Given that the test returns a positive result 90% of the time for those with the disease, we have Pr(B | A) = 0.9 (or 90%).
Step 3: Calculate Pr(B | A')
Since we know that the test returns a negative result 75% of the time for those without the disease, we can use the complement rule to find Pr(B | A') as follows:
Pr(B | A') = 1 - Pr(~B | A') [~B denotes the complement of B]
Pr(B | A') = 1 - 0.75 = 0.25 (or 25%)
Step 4: Calculate Pr(A')
To find Pr(A'), we can use the complement rule once again:
Pr(A') = 1 - Pr(A) = 1 - 0.2 = 0.8 (or 80%)
Step 5: Apply Bayes' Theorem
Now, we can use Bayes' Theorem to calculate Pr(B | A'), given the information we have:
Pr(B | A') = (Pr(B) * Pr(A' | B)) / Pr(A')
= (Pr(B) * (1 - Pr(~B | A'))) / Pr(A')
= (Pr(B) * (1 - 0.75)) / 0.8
= (Pr(B) * 0.25) / 0.8
Step 6: Calculate Pr(B)
To calculate Pr(B), we can use the law of total probability:
Pr(B) = Pr(B | A) * Pr(A) + Pr(B | A') * Pr(A')
= 0.9 * 0.2 + 0.25 * 0.8
= 0.18 + 0.2
= 0.38 (or 38%)
Finally, substitute the value of Pr(B) into the equation obtained in Step 5:
Pr(B | A') = (0.38 * 0.25) / 0.8
= 0.095 / 0.8
≈ 0.119 (or 11.9%)
Therefore, the probability of testing positive given that there is no disease is approximately 0.119 (or 11.9%).