In 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 calculate the percentage of women age 20 and younger who are or have been pregnant, we can multiply the percentage of women who have taken a pregnancy test (50%) by the percentage of those who have taken a test and are or have been pregnant (30%):
Percentage of women who are or have been pregnant = 50% * 30% = 15%
Therefore, 15% of women age 20 and younger are or have been pregnant.
2) The probability of a false negative test can be found by subtracting the true positive rate from 1. In this case, the false negative rate is given as 13%. We can calculate the probability of a false negative test as follows:
Probability of a false negative = 1 - Probability of a true positive
Probability of a false negative = 1 - 30% = 70%
Therefore, the probability that a woman had a false negative test is 70%.
3) To determine the probability that a woman who received a negative pregnancy test is actually pregnant, we need to use Bayes' theorem. Bayes' theorem states that:
P(A|B) = (P(B|A) * P(A)) / P(B)
In this case, A represents the event of being pregnant and B represents the event of receiving a negative pregnancy test.
P(B|A), the probability of receiving a negative test given that the woman is pregnant, is given as the false negative rate: 13% or 0.13.
P(A), the prior probability of being pregnant, is the percentage of women who are or have been pregnant (15% or 0.15).
P(B), the probability of receiving a negative pregnancy test, is calculated by subtracting the true positive rate from 1: 1 - 30% = 70% or 0.70.
Using Bayes' theorem,
P(A|B) = (P(B|A) * P(A)) / P(B)
P(A|B) = (0.13 * 0.15) / 0.70
P(A|B) ≈ 0.0286
Therefore, the probability that a woman who received a negative pregnancy test is actually pregnant is approximately 0.0286, or 2.86%.
4) Similarly, to determine the probability that a woman who received a positive pregnancy test is not actually pregnant, we can use Bayes' theorem. Let's denote A as the event of being pregnant and B as the event of receiving a positive pregnancy test.
P(B|~A), the probability of receiving a positive test given that the woman is not pregnant, is the false positive rate: 5% or 0.05.
P(~A), the prior probability of not being pregnant, is the complement of the percentage of women who are or have been pregnant: 1 - 15% = 85% or 0.85.
P(B), the probability of receiving a positive pregnancy test, is the true positive rate: 30% or 0.30.
Using Bayes' theorem,
P(~A|B) = (P(B|~A) * P(~A)) / P(B)
P(~A|B) = (0.05 * 0.85) / 0.30
P(~A|B) ≈ 0.1417
Therefore, the probability that a woman who received a positive pregnancy test is not actually pregnant is approximately 0.1417, or 14.17%.