In a city, a rare disease occurs in 2% of the population, but certain symptoms are fairly common (10%) due to allergies. It is known that 90% of people with the rare disease exhibit those symptoms.Based on this information, what is the probability that a person has the rare disease when they exhibit those symptoms?
Question
In a city, a rare disease occurs in 2% of the population, but certain symptoms are fairly common (10%) due to allergies. It is known that 90% of people with the rare disease exhibit those symptoms.Based on this information, what is the probability that a person has the rare disease when they exhibit those symptoms?
Solution
To solve this problem, we can use Bayes' theorem, which is a principle in probability theory and statistics that describes how to update the probabilities of hypotheses when given evidence. It's often used in medical testing.
The theorem is usually written as follows:
P(A|B) = [P(B|A) * P(A)] / P(B)
Where:
- P(A|B) is the probability of event A given event B is true.
- P(B|A) is the probability of event B given event A is true.
- P(A) and P(B) are the probabilities of events A and B respectively.
In this case:
- Event A is having the disease.
- Event B is exhibiting the symptoms.
From the problem, we know that:
- P(A), the probability of having the disease, is 2% or 0.02.
- P(B), the probability of exhibiting the symptoms, is 10% or 0.1.
- P(B|A), the probability of exhibiting the symptoms given that you have the disease, is 90% or 0.9.
We want to find P(A|B), the probability of having the disease given that you exhibit the symptoms.
Substituting these values into Bayes' theorem gives us:
P(A|B) = [P(B|A) * P(A)] / P(B) = [0.9 * 0.02] / 0.1 = 0.018
So, the probability that a person has the rare disease when they exhibit those symptoms is 0.018 or 1.8%.
Similar Questions
It is known that a rare disease affects 1% of the population. A medical test for this disease is 99% effective, which means that if you have the disease, there is a 99% chance that the test will be positive, and if you do not have the disease, there is a 99% chance that the test will be negative.If you take the medical test and result is positive, what is the chance that you have the disease?Hint: consider a cohort of 10000 people and calculate P(having the disease AND test is positive) and P(not having the disease AND test is positive)Group of answer choices0.750.990.010.5
Suppose that a certain diagnostic screening programme has the probability of 0.023 ofmissing a positive diagnosis. In a sample of 100 cases independent of each other, whatis the probability that(i) Exactly 3 cases were undiagnosed?(ii) At least 2 cases were undiagnosed?
If the probability of asthma in a patient is 10%, then the odds of asthma are:(Round your answer to 2 decimal places.).1.11.91SkipSubmit
You are studying a rare disease, which affects 1 out of 10,000 individuals. The disease is caused by a gene with two alleles: D and d. Individuals with a dd genotype are affected, while individuals with a Dd phenotype are unaffected carriers. What is the expected frequency of unaffected carriers in the population, assuming this gene is in Hardy-Weinberg equilibrium?Multiple choice question.approximately 0.02%approximately 1%approximately 2%approximately 98%
A recent study has shown that a new treatment cures a certain disease 68% of the time. If this percentage is correct, what is the probability that, in a random sample of 7 patients undergoing this treatment, exactly 4 are cured?Round your answer to three decimal places.
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.