A common blood test performed on pregnant women to screen for chromosome abnormalities in the fetus measures the human chorionic gonadotropin (hCG) hormone. Suppose that in a given population, 6% of fetuses have a chromosome abnormality. The test correctly produces a positive result for a fetus with a chromosome abnormality 88% of the time and correctly produces a negative result for a normal fetus 81% of the time. What proportion of women who get a positive test result are actually carrying a fetus with a chromosome abnormality
To determine the proportion of women who get a positive test result and are actually carrying a fetus with a chromosome abnormality, we can use Bayes' theorem.
Let's denote the following events:
A = Fetus has a chromosome abnormality
B = Positive test result
We need to find P(A|B), which represents the probability of having a chromosome abnormality given a positive test result.
Using Bayes' theorem, we have:
P(A|B) = (P(B|A) * P(A)) / P(B)
Given:
P(A) = 0.06 (probability of a fetus having a chromosome abnormality)
P(B|A) = 0.88 (probability of a positive test result given a chromosome abnormality)
To calculate P(B), we need to consider both true positive and false positive results:
P(B) = P(B|A) * P(A) + P(B|A') * P(A')
P(B|A') represents the probability of a positive test result given a normal fetus. In this case, it would be the complement of the probability of a negative test result, which is 1 - 0.81 = 0.19.
P(A') is the complement of P(A), so P(A') = 1 - P(A) = 1 - 0.06 = 0.94.
Now we can calculate P(B):
P(B) = (0.88 * 0.06) + (0.19 * 0.94) = 0.0528 + 0.1786 = 0.2314
Finally, we can substitute the values into the formula to find P(A|B):
P(A|B) = (0.88 * 0.06) / 0.2314 = 0.0528 / 0.2314 ≈ 0.2284
Therefore, approximately 22.84% of women who get a positive test result are actually carrying a fetus with a chromosome abnormality.
To find the proportion of women who get a positive test result and actually carry a fetus with a chromosome abnormality, we can use Bayes' theorem. Bayes' theorem allows us to update the probability of an event based on new information.
Let's break down the information given:
1. The prevalence of chromosome abnormalities in the population is 6%.
2. The test correctly produces a positive result for a fetus with a chromosome abnormality 88% of the time (also known as sensitivity).
3. The test correctly produces a negative result for a normal fetus 81% of the time (also known as specificity).
To find the proportion of women who get a positive test result and actually carry a fetus with a chromosome abnormality, we need to find the conditional probability of having a chromosome abnormality given a positive test result.
Let's define the following events:
A = Carrying a fetus with a chromosome abnormality
B = Receiving a positive test result
We need to find P(A|B), which is the probability of having a chromosome abnormality given a positive test result.
Using Bayes' theorem, we can write:
P(A|B) = (P(B|A) * P(A)) / P(B)
Here's how to calculate each component:
P(B|A) = Sensitivity = 88% = 0.88
P(A) = Prevalence = 6% = 0.06
P(B) = (P(B|A) * P(A)) + (P(B|~A) * P(~A))
To calculate P(B), we need to consider the probability of getting a positive result when you have a chromosome abnormality (P(B|A)) and the probability of getting a positive result when you do not have a chromosome abnormality (P(B|~A)). Additionally, we need to consider the probability of having a chromosome abnormality (P(A)) and the probability of not having a chromosome abnormality (P(~A)).
P(B|~A) = (1 - Specificity) = (1 - 0.81) = 0.19
P(~A) = 1 - P(A) = 1 - 0.06 = 0.94
Now, we can substitute these values back into the equation:
P(B) = (P(B|A) * P(A)) + (P(B|~A) * P(~A))
= (0.88 *0.06) + (0.19 * 0.94)
= 0.0528 + 0.1786
= 0.2314
Lastly, substitute the values of P(B|A), P(A), and P(B) back into the original equation to find P(A|B):
P(A|B) = (P(B|A) * P(A)) / P(B)
= (0.88 *0.06) / 0.2314
= 0.0528 / 0.2314
≈ 0.2283
Therefore, approximately 22.83% of women who receive a positive test result are actually carrying a fetus with a chromosome abnormality.