To ascertain the effectiveness of a new, quicker, and simpler test for identifying diabetes, a study was undertaken involving 739 participants. With this test, a positive result indicates the presence of diabetes. A negative result indicates the absence of diabetes. Data collected to assess the effectiveness of this new test revealed: 201 of the participants confirmed they actually had diabetes, with the other participants confirming they actually did not have diabetes. 124 participants actually had diabetes and produced a negative test result. 255 participants produced a negative test result. A participant from the study is chosen at random. What is the probability they do actually have diabetes given the test result is positive? (3 decimal places)
Question
To ascertain the effectiveness of a new, quicker, and simpler test for identifying diabetes, a study was undertaken involving 739 participants. With this test, a positive result indicates the presence of diabetes. A negative result indicates the absence of diabetes. Data collected to assess the effectiveness of this new test revealed:
201 of the participants confirmed they actually had diabetes, with the other participants confirming they actually did not have diabetes. 124 participants actually had diabetes and produced a negative test result. 255 participants produced a negative test result. A participant from the study is chosen at random. What is the probability they do actually have diabetes given the test result is positive? (3 decimal places)
Solution
To solve this problem, we need to use the concept of conditional probability.
First, let's define the events:
Event D: The participant has diabetes. Event T: The test result is positive.
We are asked to find P(D|T), the probability that a participant has diabetes given that the test result is positive.
From the problem, we know:
- 201 participants actually have diabetes.
- 124 participants have diabetes and produced a negative test result, so 201 - 124 = 77 participants have diabetes and produced a positive test result.
- 255 participants produced a negative test result, so 739 - 255 = 484 participants produced a positive test result.
Now we can calculate P(D|T) using the formula:
P(D|T) = P(D ∩ T) / P(T)
P(D ∩ T) is the probability that a participant has diabetes and the test result is positive, which is 77/739.
P(T) is the probability that the test result is positive, which is 484/739.
So,
P(D|T) = (77/739) / (484/739) = 77 / 484 = 0.159
So, the probability that a participant has diabetes given that the test result is positive is approximately 0.159 or 15.9% when rounded to three decimal places.
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