A 2020 statistic shows that 50% of women age 20 and younger have taken a pregnancy test and 30% of those who have taken a pregnancy test are or have been pregnant. Over the counter pregnancy tests give a false positive 5% of the time and a false negative 13% of the time.
1). Assuming that none of the women who did not take a pregnancy test were pregnant, what percentage of women age 20 and under are or have been pregnant?
2) Given that a woman took a pregnancy test, what is the probability that she had a false negative test?
3) For a woman who received a negative pregnancy test, what is the probability she actually was pregnant?
4) For a woman who received a positive pregnancy test, what is the probability that she was not pregnant?
1) To determine the percentage of women age 20 and under who are or have been pregnant, we can multiply the probabilities of two events: taking a pregnancy test and being pregnant after taking the test.
The percentage of women who have taken a pregnancy test is 50% (given in the question). Among those who have taken the test, the percentage of them who are or have been pregnant is 30%. To find the overall percentage of women who are or have been pregnant, we multiply these probabilities:
Percentage of women who are or have been pregnant = 50% * 30% = 15%
Therefore, 15% of women age 20 and under are or have been pregnant based on the given statistics.
2) To find the probability of a false negative test, we need to use Bayes' theorem. Bayes' theorem states that the probability of an event A given event B is equal to the probability of event A and event B happening together, divided by the probability of event B.
Let's denote FNT as a false negative test and TN as a true negative:
Probability of a false negative test = (Probability of a false negative test and being pregnant) / (Probability of being pregnant)
From the given information, we know that 13% of pregnancy tests are false negatives (FNT). We also know that 30% of women who have taken the test are or have been pregnant. So the probability of a false negative test and being pregnant is 13% * 30% = 3.9%.
Now, the probability of being pregnant is already given as 30%, so we can substitute the values into the formula:
Probability of a false negative test = 3.9% / 30% = 13%
Therefore, the probability that a woman who took a pregnancy test had a false negative result is 13%.
3) To find the probability that a woman who received a negative pregnancy test is actually pregnant, we can use Bayes' theorem again:
Probability of being pregnant given a negative test = (Probability of being pregnant and a negative test) / (Probability of a negative test)
Let's denote PNT as a true negative test:
Probability of being pregnant given a negative test = (Probability of being pregnant and a negative test) / (Probability of a negative test)
From the given information, we know that false negatives account for 13% of the negative tests and true negatives account for the remaining percentage.
Probability of being pregnant and a negative test = (Percentage of false negatives) * (Percentage of women who have taken a test) = 13% * 50% = 6.5%
Now, the percentage of negative tests is equal to the true negative tests:
Probability of a negative test = Percentage of true negatives = 100% - Percentage of false positives = 100% - 5% = 95%
Substituting the values into the formula:
Probability of being pregnant given a negative test = 6.5% / 95% ≈ 0.0684 or 6.84%
Therefore, the probability that a woman who received a negative pregnancy test is actually pregnant is approximately 6.84%.
4) To find the probability that a woman who received a positive pregnancy test is not pregnant, we can again use Bayes' theorem:
Probability of not being pregnant given a positive test = (Probability of not being pregnant and a positive test) / (Probability of a positive test)
Let's denote FPT as a false positive test and TP as a true positive:
Probability of not being pregnant and a positive test = (Probability of not being pregnant) * (Probability of a false positive test) = (100% - Percentage of being pregnant) * (Percentage of false positives) = (100% - 30%) * 5% = 70% * 5% = 3.5%
The probability of a positive test is equal to the true positives:
Probability of a positive test = Percentage of true positives = Percentage of being pregnant = 30%
Substituting the values into the formula:
Probability of not being pregnant given a positive test = 3.5% / 30% ≈ 0.1167 or 11.67%
Therefore, the probability that a woman who received a positive pregnancy test is not pregnant is approximately 11.67%.