Pregnancy Test Results
Positive Test Result
(Pregnancy is indicated)
Negative Test Result
(Pregnancy is not indicated)
Subject is pregnant
80
5
Subject is not pregnant
3
11
a) Based on the results in the table, what is the probability of a women being pregnant if the test
indicates a negative result? If you are a physician and you have a patient who tested negative
what advice would you give?
b) Based on the results in the table, what is the probability of a false positive? That is, what is the
probability of getting a positive result if the woman is not actually pregnant? If you are a
physician, and you have a patient who tested positive, what advice would you give?
c) Find the values of each of the following, and explain the difference between the two events.
Describe the concept confusion of the inverse in this context.
i. P(pregnant | positive test result)
ii. P(positive test result | pregnant)
a) To find the probability of a woman being pregnant if the test indicates a negative result, we need to use Bayes' theorem:
P(pregnant | negative test result) = (P(negative test result | pregnant) * P(pregnant)) / P(negative test result)
From the table, we can see that out of a total of 8 subjects who tested negative, 3 are not pregnant. Therefore, the probability of a negative test result given that the woman is pregnant is 3/8. Assuming an equal probability of being pregnant or not being pregnant, P(pregnant) would be 8/19 (80 divided by the sum of all subjects' results: 80+5+3+11).
Now we need to find P(negative test result), which we can calculate by subtracting the probability of a positive test result from 1. From the table, the total number of positive test results is 5+80=85 out of 108 cases, so P(negative test result) would be 1 - 85/108.
Finally, we can calculate the probability:
P(pregnant | negative test result) = (3/8 * 8/19) / (1 - 85/108)
As a physician, if you have a patient who tested negative, the probability of her being pregnant is given by this calculated value. Based on this probability, you should consider additional tests or consult with the patient to gather more information before making any definitive recommendations.
b) To find the probability of a false positive, we need to calculate:
P(positive test result | not pregnant) = (P(positive test result & not pregnant) / P(not pregnant)
From the table, there are 5 subjects who tested positive but are not pregnant. Out of the total number of subjects who are not pregnant (3+11=14), the probability would be 5/14.
As a physician, if you have a patient who tested positive, the probability of it being a false positive is given by the calculated value. Based on this probability, you may consider confirming the result with additional tests or discussing the possibility of false positives with the patient.
c) In this context, the events can be defined as follows:
i. P(pregnant | positive test result): The probability of a woman being pregnant given a positive test result.
ii. P(positive test result | pregnant): The probability of getting a positive test result given that the woman is pregnant.
The difference between these two events is the order in which they are presented. Event i focuses on the probability of pregnancy given a positive test result, whereas event ii focuses on the probability of getting a positive test result given that the woman is already pregnant.
The concept confusion of the inverse in this context refers to the fact that these two events are not the same. The probability of being pregnant given a positive result may not be the same as the probability of getting a positive result given that the woman is pregnant. This confusion arises due to the different conditional probabilities involved. It is important to carefully consider the order of events and the conditional probabilities associated with them to avoid misinterpretation.