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?
1) To determine the percentage of women age 20 and under who are or have been pregnant, we need to consider two groups: those who took a pregnancy test and those who did not.
Let's define the following variables:
A = women who are or have been pregnant
B = women who took a pregnancy test
Based on the given statistics, we know that 50% of women age 20 and under have taken a pregnancy test. Therefore, the percentage of women who took a pregnancy test is 50%.
Now, we need to find the percentage of women who are or have been pregnant among those who took a pregnancy test. We know that 30% of those who took a pregnancy test are or have been pregnant. So, the percentage of women who are or have been pregnant and took a pregnancy test is 30% of 50%, which is 0.30 * 0.50 = 0.15 (or 15%).
To find the total percentage of women age 20 and under who are or have been pregnant, we sum the percentages from both groups:
- Women who are or have been pregnant and took a pregnancy test: 15%
- Women who are or have been pregnant and did not take a pregnancy test: 0% (since none of the women who did not take a pregnancy test were pregnant)
Adding these percentages together, we get:
15% + 0% = 15%
Therefore, the percentage of women age 20 and under who are or have been pregnant is 15%.
2) To find the probability that a woman had a false negative test given that she took a pregnancy test, we need to use conditional probability.
Let's define the following variables:
C = a false negative pregnancy test
B = a woman took a pregnancy test
We know that a false negative occurs with a probability of 13% (0.13). So, the probability that a woman had a false negative test given that she took a pregnancy test is equal to the probability of C happening when B happened: P(C|B).
We can calculate P(C|B) using Bayes' theorem:
P(C|B) = P(C and B) / P(B)
We know that a false negative occurs with a probability of 13% (0.13) and 50% of women age 20 and under took a pregnancy test. So, P(C and B) = 0.13 * 0.50 = 0.065 (or 6.5%).
Similarly, P(B) = 0.50 (or 50%).
Therefore, the probability that a woman had a false negative test given that she took a pregnancy test is:
P(C|B) = P(C and B) / P(B) = 0.065 / 0.50 = 0.13 (or 13%).
So, the probability that a woman had a false negative test given that she took a pregnancy test is 13%.
3) To find the probability that a woman who received a negative pregnancy test was actually pregnant, we need to use conditional probability.
Let's define the following variables:
D = a woman is actually pregnant
E = a woman received a negative pregnancy test
We know that a false negative occurs with a probability of 13% (0.13). Therefore, the probability that a woman received a negative pregnancy test when she was actually pregnant is 1 - 0.13 = 0.87 (or 87%).
To find the probability that a woman who received a negative pregnancy test was actually pregnant, we need to calculate P(D|E) using Bayes' theorem:
P(D|E) = P(D and E) / P(E)
We know that the probability of a woman being actually pregnant and receiving a negative pregnancy test is the product of the probability of being actually pregnant (30% or 0.30) and the probability of receiving a negative test given that she was actually pregnant (87% or 0.87). So, P(D and E) = 0.30 * 0.87 = 0.261 (or 26.1%).
The probability of receiving a negative pregnancy test is the sum of the probabilities of receiving a true negative (87%) and a false negative (13%). So, P(E) = 0.87 + 0.13 = 1.
Therefore, the probability that a woman who received a negative pregnancy test was actually pregnant is:
P(D|E) = P(D and E) / P(E) = 0.261 / 1 = 0.261 (or 26.1%).
Thus, the probability that a woman who received a negative pregnancy test was actually pregnant is 26.1%.
4) To find the probability that a woman who received a positive pregnancy test was not pregnant, we need to use conditional probability.
Let's define the following variables:
F = a woman is not pregnant
G = a woman received a positive pregnancy test
We know that a false positive occurs with a probability of 5% (0.05). Therefore, the probability that a woman received a positive pregnancy test when she was not actually pregnant is 1 - 0.05 = 0.95 (or 95%).
To find the probability that a woman who received a positive pregnancy test was not pregnant, we need to calculate P(F|G) using Bayes' theorem:
P(F|G) = P(F and G) / P(G)
We know that the probability of a woman being not pregnant and receiving a positive pregnancy test is the product of the probability of being not pregnant (70% or 0.70) and the probability of receiving a positive test given that she was not pregnant (95% or 0.95). So, P(F and G) = 0.70 * 0.95 = 0.665 (or 66.5%).
The probability of receiving a positive pregnancy test is the sum of the probabilities of receiving a true positive (95%) and a false positive (5%). So, P(G) = 0.95 + 0.05 = 1.
Therefore, the probability that a woman who received a positive pregnancy test was not pregnant is:
P(F|G) = P(F and G) / P(G) = 0.665 / 1 = 0.665 (or 66.5%).
Hence, the probability that a woman who received a positive pregnancy test was not pregnant is 66.5%.