Help please
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 can use the concept of conditional probability and Bayes' theorem.
1) To find the percentage of women age 20 and under who are or have been pregnant, we need to calculate the joint probability of two events: being pregnant and taking a pregnancy test.
Let's denote:
P(P) = probability of being pregnant
P(T) = probability of taking a pregnancy test
P(P|T) = probability of being pregnant given that a pregnancy test is taken
Given information:
P(T|P) = 0.3 (probability of taking a pregnancy test given that a woman is pregnant)
P(T|~P) = 0.5 (probability of taking a pregnancy test given that a woman is not pregnant)
We are also given:
P(FP) = 0.05 (probability of a false positive)
P(FN) = 0.13 (probability of a false negative)
Using Bayes' theorem, we have:
P(P|T) = (P(T|P) * P(P)) / P(T)
Since all women either take a pregnancy test or don't, we can write:
P(T) = P(T|P) * P(P) + P(T|~P) * P(~P)
= 0.3 * P(P) + 0.5 * P(~P)
Now we can substitute these values into the equation:
P(P|T) = (0.3 * P(P)) / (0.3 * P(P) + 0.5 * P(~P))
Assuming none of the women who did not take a pregnancy test were pregnant, P(~P) = 1 - P(P)
Substituting this into the equation:
P(P|T) = (0.3 * P(P)) / (0.3 * P(P) + 0.5 * (1 - P(P)))
2) To find the probability of a false negative given that a woman took a pregnancy test, we can use Bayes' theorem again.
Let's denote:
P(FN|T) = probability of a false negative given that a woman took a pregnancy test
Using Bayes' theorem:
P(FN|T) = (P(T|FN) * P(FN)) / P(T)
Since we are given that the false negative rate is 13%, we have:
P(FN) = 0.13
The probability of taking a pregnancy test given a false negative is 1 (since if there is a false negative, it means the woman definitely took the test).
Now we need to calculate P(T) as mentioned above.
Substituting these values into the equation:
P(FN|T) = (1 * 0.13) / P(T)