In a 2020 statistics show 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?
To answer these questions, we'll need to use conditional probability and apply Bayes' theorem. Let's go through each question one by one:
1) Assuming that none of the women who did not take a pregnancy test were pregnant, we need to find the percentage of women who are or have been pregnant among those who took a pregnancy test. From the given statistics, we know 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.
To calculate the percentage of women who are or have been pregnant, we multiply the probability of taking a pregnancy test by the probability of being pregnant given that a test was taken:
Percentage of women who are or have been pregnant = Probability of taking a pregnancy test * Probability of being pregnant given a test was taken
= 0.5 * 0.3
= 0.15
Therefore, 15% of women age 20 and younger are or have been pregnant, assuming that none of the women who did not take a pregnancy test were pregnant.
2) Given that a woman took a pregnancy test, we want to find the probability that she had a false negative test. From the given statistics, we know that over the counter pregnancy tests give a false negative 13% of the time.
To find the probability of a false negative test, we need to consider the percentage of women who took a pregnancy test and were pregnant, as well as the percentage of women who took a pregnancy test and were not pregnant. Let's assume there are 100 women who took a pregnancy test. Out of these 100 women, 30% (since 30% of those who took the test are pregnant) would be pregnant, and the remaining 70% would not be pregnant.
Since a false negative test means the test incorrectly shows a negative result when a woman is actually pregnant, the probability of a false negative would be equal to the percentage of women who took a pregnancy test and were pregnant but erroneously received a negative result. From the given information, we know that the false negative rate is 13%.
Therefore, the probability of a false negative test would be:
Probability of false negative = Probability of being pregnant * Probability of a false negative
= 0.3 * 0.13
= 0.039
So, the probability that a woman who took a pregnancy test had a false negative is 3.9%.
3) For a woman who received a negative pregnancy test, this question is asking about the probability that she was actually pregnant. To find this probability, we can use Bayes' theorem, which states:
Probability of A given B = (Probability of B given A * Probability of A) / Probability of B
In this case, event A represents receiving a negative pregnancy test, and event B represents being pregnant.
From the given statistics, we know that over the counter pregnancy tests give a false negative 13% of the time and a false positive 5% of the time. So, the probability of a negative test correctly indicating a non-pregnancy is 1 - 0.13 (false negative rate) = 0.87.
Using Bayes' theorem:
Probability of being pregnant given a negative test = (Probability of a negative test given being pregnant * Probability of being pregnant) / Probability of a negative test indicating non-pregnancy
= (1 - 0.13) * 0.3 / (1 - Probability of a false positive)
= 0.87 * 0.3 / (1 - 0.05)
= 0.261 / 0.95
= 0.2747
Therefore, the probability that a woman who received a negative pregnancy test is actually pregnant is approximately 27.47%.
4) For a woman who received a positive pregnancy test, the question asks for the probability that she was not pregnant. Similar to question 3, we will use Bayes' theorem.
Using Bayes' theorem:
Probability of not being pregnant given a positive test = (Probability of a positive test given not being pregnant * Probability of not being pregnant) / Probability of a positive test
The probability of a positive test given not being pregnant is equal to 1 - false positive rate, which is 1 - 0.05 = 0.95.
Using Bayes' theorem:
Probability of not being pregnant given a positive test = (0.95 * (1 - 0.3)) / (0.95 * (1 - 0.3) + 0.05 * 0.3)
= (0.95 * 0.7) / (0.95 * 0.7 + 0.015)
= 0.665 / 0.665 + 0.015
= 0.665 / 0.68
≈ 0.9779
Therefore, the probability that a woman who received a positive pregnancy test is not pregnant is approximately 97.79%.