2020 statistics 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 answer these questions, we can use conditional probability and Bayes' theorem. Let's break down each question step by step:
1) To find the percentage of women age 20 and under who are or have been pregnant, we need to consider both the percentage of women who have taken a pregnancy test and the percentage of those who have been pregnant.
According to the given information, 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 find the overall percentage, we multiply these two probabilities: 50% * 30% = 15%. Therefore, 15% of women age 20 and under are or have been pregnant.
2) To determine the probability of a false negative test, we need to consider the percentage of false negatives relative to the total number of women who took a pregnancy test.
According to the given information, false negatives occur 13% of the time. Therefore, the probability of a false negative test is 13%.
3) To find the probability that a woman who received a negative pregnancy test was actually pregnant, we need to use Bayes' theorem.
Let's define the events:
A: Woman is pregnant
B: Woman receives a negative pregnancy test
We want to find P(A|B), the probability that a woman is pregnant given that she received a negative pregnancy test.
Bayes' theorem states:
P(A|B) = (P(B|A) * P(A)) / P(B)
We know that P(B|A) is the probability of a false negative, which is given as 13%. P(A) is the overall percentage of women who are or have been pregnant, which is 15%.
To find P(B), the probability of receiving a negative pregnancy test, we subtract the probability of a positive pregnancy test (complement) from 1.
Given that false positives occur 5% of the time, the probability of a positive pregnancy test is 5%.
Therefore, P(B) = 1 - 5% = 95%.
Now we can calculate P(A|B) using Bayes' theorem:
P(A|B) = (P(B|A) * P(A)) / P(B)
= (13% * 15%) / 95%
= 0.0204 or 2.04%
Therefore, the probability that a woman who received a negative pregnancy test was actually pregnant is 2.04%.
4) To find the probability that a woman who received a positive pregnancy test was not pregnant, we can use similar reasoning as in the previous question.
Again, let's define the events:
A: Woman is pregnant
B: Woman receives a positive pregnancy test
We want to find P(~A|B), the probability that a woman is not pregnant given that she received a positive pregnancy test.
Using Bayes' theorem, we have:
P(~A|B) = (P(B|~A) * P(~A)) / P(B)
We know that P(B|~A) is the probability of a false positive, given as 5%. P(~A) is the complement of P(A), which is 100% - 15% = 85%.
P(B) remains the same, at 95%.
Now we can calculate P(~A|B) using Bayes' theorem:
P(~A|B) = (P(B|~A) * P(~A)) / P(B)
= (5% * 85%) / 95%
= 0.0447 or 4.47%
Therefore, the probability that a woman who received a positive pregnancy test was not pregnant is 4.47%.