does anybody know probability
Thank you in advance
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?
To answer these questions, we can use the concepts of conditional probability and Bayes' theorem. Let's break down each question and explain how to get the answers.
1) To find the percentage of women age 20 and under who are or have been pregnant, we need to consider both those who have taken a pregnancy test and those who have not. Since the statistic given only provides information about those who have taken a pregnancy test, we need to make some assumptions.
Assuming none of the women who did not take a pregnancy test were pregnant, we can calculate the overall probability by multiplying the probability of taking a test (50%) with the probability of being pregnant given a positive test (30%).
Percentage of women age 20 and under who are or have been pregnant = Probability of taking a test * Probability of being pregnant given a positive test
= 0.5 * 0.3
= 0.15 or 15%
Therefore, the percentage of women age 20 and under who are or have been pregnant is 15%.
2) The probability that a woman had a false negative test can be determined by using the information provided about false negatives. The false negative rate given in the question is 13%.
Probability of a false negative = 13%
Therefore, the probability that a woman had a false negative test is 13%.
3) For a woman who received a negative pregnancy test, we need to use Bayes' theorem to calculate the probability that she actually was pregnant. Bayes' theorem states:
P(A|B) = (P(B|A) * P(A)) / P(B)
Let's use A to represent the event "woman is pregnant" and B to represent the event "woman received a negative pregnancy test".
P(A) is the probability of being pregnant, which we determined in question 1 to be 15%.
P(B|A) is the probability of receiving a negative test given that the woman is pregnant. This is the complement of the probability of a false positive, which is 1 - 5% = 95%.
P(B) is the probability of receiving a negative test, which includes both the cases where the woman is pregnant but the test gave a false negative, and the cases where the woman is not pregnant and the test gave a true negative. So, it can be calculated as:
P(B) = (Probability of being pregnant * Probability of a false negative) + (Probability of not being pregnant * Probability of a true negative)
P(B) = (0.15 * 0.13) + (0.85 * 0.87)
Now, we can plug these values into Bayes' theorem:
P(A|B) = (P(B|A) * P(A)) / P(B)
P(A|B) = (0.95 * 0.15) / P(B)
Calculate the value of P(B) using the above equation and then substitute it in the equation to find P(A|B).
4) For a woman who received a positive pregnancy test, we can similarly use Bayes' theorem to calculate the probability that she was not pregnant.
P(A) is the probability of being pregnant, which we determined in question 1 to be 15%.
P(B|A) is the probability of receiving a positive test given that the woman is pregnant. This is the true positive rate, which is 30%.
P(B) is the probability of receiving a positive test, which includes both the cases where the woman is pregnant and the test gave a true positive, and the cases where the woman is not pregnant but the test gave a false positive. So, it can be calculated as:
P(B) = (Probability of being pregnant * Probability of a true positive) + (Probability of not being pregnant * Probability of a false positive)
P(B) = (0.15 * 0.3) + (0.85 * 0.05)
Now, we can plug these values into Bayes' theorem:
P(A|B) = (P(B|A) * P(A)) / P(B)
P(A|B) = (0.3 * 0.15) / P(B)
Calculate the value of P(B) using the above equation and then substitute it in the equation to find P(A|B).
By following these steps, you can calculate the probabilities required to answer each question.