Can someone please help me? certain medical diagnostic test is used to determine if people have pinkyitis and it is known that 1 in 1300 Americans has the disease. The test correctly diagnoses the presence of pinkyitis 97% of the time and it correctly diagnoses the absence of the disease 98% of the time. Find the probability that a person has the disease given that the test says that he has the disease. Find the probability that a person does not have the disease given that the test says that he does not have the disease
To find the probability that a person has the disease given that the test says they have the disease, we can use Bayes' theorem:
P(A|B) = (P(B|A) * P(A)) / P(B)
Let's define the events:
A = person has the disease
B = test says person has the disease
Given information:
P(A) = 1 in 1300 = 1/1300
P(B|A) = the test correctly diagnoses the presence of pinkyitis = 97% = 0.97
Now we need to find P(B), the probability that the test says a person has the disease. To do this, we need to consider both true positives (people with the disease who test positive) and false positives (people without the disease who test positive).
P(B) = (P(B|A) * P(A)) + (P(B|not A) * P(not A))
We know P(B|A) = 0.97 and P(A) = 1/1300. We also know that only 1 in 1300 Americans has the disease, so P(not A) = 1 - P(A) = 1 - (1/1300).
Now we need to find P(B|not A), which is the probability that the test incorrectly says a person has the disease when they do not. This can be calculated as 1 - P(not B|not A), where P(not B|not A) is the probability that the test correctly diagnoses the absence of the disease. Given that the test correctly diagnoses the absence of the disease 98% of the time, P(not B|not A) = 0.98.
Finally, we can calculate P(B):
P(B) = (0.97 * 1/1300) + (0.98 * (1 - (1/1300)))
Now we can substitute these values into the formula to calculate P(A|B), the probability that a person has the disease given that the test says they have the disease:
P(A|B) = (P(B|A) * P(A)) / P(B)
P(A|B) = (0.97 * 1/1300) / P(B)
Similarly, to find the probability that a person does not have the disease given that the test says they do not have the disease, we can use the same approach:
P(not A|not B) = (P(not B|not A) * P(not A)) / P(not B)
Here, P(not B) can be calculated as 1 - P(B).
Please substitute the calculated values to find the final probabilities.