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?
To solve these questions, we can use conditional probability and Bayes' theorem. Let's break down each question and explain how to calculate the probabilities step by step:
1) The percentage of women age 20 and under who are or have been pregnant can be calculated using the concept of conditional probability. We know that 50% of women in this age group have taken a pregnancy test, and out of those who took the test, 30% were or have been pregnant.
To calculate the overall percentage of women in this age group who are or have been pregnant, we multiply the two probabilities together:
Probability of a woman being or having been pregnant = Probability of taking a pregnancy test * Probability of being or having been pregnant given a pregnancy test
Pregnant = 0.5 * 0.3 = 0.15 (15%)
Therefore, 15% of women age 20 and under are or have been pregnant.
2) The probability of a false negative test can be calculated using conditional probability. We know that 13% of over the counter pregnancy tests give a false negative result.
To calculate the probability that a woman who took a pregnancy test had a false negative, we use the formula:
Probability of a false negative = Probability of a false negative * Probability of taking a pregnancy test
False Negative = 0.13 * 0.5 = 0.065 (6.5%)
Therefore, the probability of a woman having a false negative test is 6.5%.
3) To calculate the probability that a woman who received a negative pregnancy test is actually pregnant, we can use Bayes' theorem.
The formula for Bayes' theorem is:
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.
We already know from previous calculations that:
P(A) = 0.15 (probability of being or having been pregnant)
P(B|A) = 0.13 (probability of a false negative given pregnancy)
P(B) = 1 - P(A) = 1 - 0.15 = 0.85 (probability of receiving a negative test)
Using Bayes' theorem:
P(A|B) = (P(B|A) * P(A)) / P(B)
Pregnant given a negative test = (0.13 * 0.15) / 0.85 = 0.023 (2.3%)
Therefore, the probability that a woman who received a negative pregnancy test is actually pregnant is 2.3%.
4) To calculate the probability that a woman who received a positive pregnancy test is not pregnant, we can again use Bayes' theorem.
Using the same notation as before:
A represents the event of being pregnant, and B represents the event of receiving a positive pregnancy test.
We already know from previous calculations that:
P(A) = 0.15 (probability of being or having been pregnant)
P(B|A) = 1 - 0.05 = 0.95 (probability of a true positive given pregnancy)
P(B) = 0.5 (probability of receiving a positive test)
Using Bayes' theorem:
P(A|B) = (P(B|A) * P(A)) / P(B)
Not Pregnant given a positive test = (0.05 * 0.85) / 0.5 = 0.085 (8.5%)
Therefore, the probability that a woman who received a positive pregnancy test is not pregnant is 8.5%.