A new medical test has been designed to detect the presence of the mysterious Brainlesserian disease. Among those who have the disease, the probability that the disease will be detected by the new test is 0.7. However, the probability that the test will erroneously indicate the presence of the disease in those who do not actually have it is 0.1. It is estimated that 20 % of the population who take this test have the disease.
If the test administered to an individual is positive, what is the probability that the person actually has the disease?
Take 10000 people, 2000 will have the disease (20 percent) and 8000 won't have the disease.
of the 2000 that have the disease, 1400 test positive (.7*2000)
of the 8000 that do not have the disease, (8000*.1) 800 will test postive
so a total of 800+1400=2200 will test postive.
but 2000 have the disease, so the probability of testing positive and having the disease is 2000/2200=.91 about.
Well, well, well, looks like we've got a case of Brainlesserian disease. But fear not, my dear human friend, as I have the perfect prescription for you - a dose of probability humor!
Let's try to dissect this tricky scenario step by step, shall we? We know that the probability of someone with the disease testing positive is 0.7. And we also know that the probability of a false positive (someone without the disease testing positive) is 0.1. Just like spotting a clown in a crowded room, things can get a bit confusing!
Now, let's say we have a group of 100 people taking the test. According to estimates, around 20% of them actually have the disease. That means we have 20 people in our group who are true positives.
But hold on, we also have to factor in the false positives, those poor people who are misdiagnosed. With a false positive rate of 0.1, that means we'll have 10 false positives in our group.
So, out of the 30 people who received a positive test result, we know that 20 of them have the disease and 10 of them do not. It's like separating the real clowns from those imposter clowns!
Now, if you're one of those 30 individuals who tested positive, what's the probability that you actually have the disease? Well, out of the 30 people, 20 of them have the disease. So, the probability of you having it would be 20 out of 30, which simplifies to 2 out of 3, or approximately 0.67.
So, my friend, there's a 0.67 chance that you've caught the Brainlesserian disease. Remember, laughter is the best medicine, so keep smiling even if you have a case of the "Brainlesserian" blues!
To determine the probability that a person actually has the disease given a positive test result, we can use Bayes' theorem.
Let's define the probabilities as follows:
P(D) = Probability of having the disease (Prior probability) = 0.20 (20%)
P(Pos|D) = Probability of a positive test result given the person has the disease (Sensitivity) = 0.70 (70%)
P(Pos|~D) = Probability of a positive test result given the person does not have the disease (Specificity) = 0.10 (10%)
We want to find P(D|Pos) which is the probability of having the disease given a positive test result.
1. Calculate the probability of a positive test result (P(Pos)):
P(Pos) = P(D) * P(Pos|D) + P(~D) * P(Pos|~D)
= 0.20 * 0.70 + 0.80 * 0.10
= 0.14 + 0.08
= 0.22
2. Apply Bayes' theorem:
P(D|Pos) = P(D) * P(Pos|D) / P(Pos)
Substituting the values:
P(D|Pos) = 0.20 * 0.70 / 0.22
= 0.14 / 0.22
≈ 0.6364
Therefore, the probability that a person actually has the disease given a positive test result is approximately 0.6364 or 63.64%.
To find the probability that the person actually has the disease given a positive test result, we can use Bayes' theorem. Bayes' theorem allows us to update our belief about the probability of an event based on new evidence.
Let's define the events:
A: The person has the Brainlesserian disease.
B: The test result is positive.
We are given the following probabilities:
P(A) = 0.2 (20% of the population have the disease)
P(B|A) = 0.7 (probability the test is positive given the person has the disease)
P(B|not A) = 0.1 (probability the test is positive given the person does not have the disease)
We want to find P(A|B), the probability that the person actually has the disease given a positive test result.
Using Bayes' theorem, we have:
P(A|B) = (P(B|A) * P(A)) / P(B)
To calculate P(B), we need to consider the probabilities of two mutually exclusive events:
1. The person has the disease and the test is positive (A and B)
2. The person does not have the disease and the test is positive (not A and B)
P(B) = P(A and B) + P(not A and B) = (P(B|A) * P(A)) + (P(B|not A) * P(not A))
Substituting the given values, we have:
P(B) = (0.7 * 0.2) + (0.1 * 0.8) = 0.14 + 0.08 = 0.22
Now we can substitute this into the original equation to find P(A|B):
P(A|B) = (0.7 * 0.2) / 0.22 = 0.14 / 0.22 ≈ 0.6364
Therefore, the probability that the person actually has the Brainlesserian disease given a positive test result is approximately 0.6364, or 63.64%.