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.
Let's denote the following:
A: Having the disease
B: Test says the person has the disease
According to the information given, the probability of having the disease, P(A), is 1 in 1300 or 1/1300.
The probability that the test says the person has the disease given that they actually have the disease, P(B|A), is 97% or 0.97. This is known as the sensitivity of the test.
Now, we can calculate the probability that the test says the person has the disease, P(B):
P(B) = P(A) * P(B|A) + P(not A) * P(B|not A)
P(not A) is the probability of not having the disease, which is equal to 1 - P(A) or 1 - 1/1300.
P(B|not A) is the probability that the test says the person has the disease given that they don't actually have it. This is known as the false positive rate and is given as 2% or 0.02.
Therefore,
P(B) = (1/1300) * (0.97) + (1 - 1/1300) * (0.02)
Next, we can find the probability that a person has the disease given that the test says they have it, P(A|B), by applying Bayes' theorem:
P(A|B) = P(A) * P(B|A) / P(B)
Substituting the values we obtained earlier,
P(A|B) = (1/1300) * (0.97) / P(B)
By calculating P(A|B), we will get the probability that a person has the disease given that the test says they have it.
Similarly, to find the probability that a person does not have the disease given that the test says they do not have it, P(not A|not B), we can use Bayes' theorem again:
P(not A|not B) = P(not A) * P(not B|not A) / P(not B)
P(not B) is the probability that the test says the person does not have the disease, given that they actually don't have it. This is known as the specificity of the test and is given as 98% or 0.98.
P(not B|not A) is the probability that the test says the person does not have the disease given that they don't actually have it, and is equal to 1 - 0.98 or 0.02.
Hence,
P(not B) = P(not A) * P(not B|not A) + P(A) * P(not B|A)
P(not A|not B) = P(not A) * P(not B|not A) / P(not B)
By calculating P(not A|not B), we will get the probability that a person does not have the disease given that the test says they do not have it.
I hope this explanation helps you understand the process of finding these probabilities. If you have the specific values for P(B), you can substitute them into the formulae to obtain the final results.