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. Bayes' theorem states that the probability of event A occurring given that event B has occurred is equal to the probability of event B occurring given that event A has occurred, multiplied by the probability of event A occurring, divided by the probability of event B occurring.

Let's denote:
P(pregnant) = probability of a woman being pregnant = 80/100 = 0.8
P(not pregnant) = probability of a woman not being pregnant = 20/100 = 0.2
P(negative test result | pregnant) = probability of a negative test result given that the woman is pregnant = 5/80 = 0.0625
P(negative test result | not pregnant) = probability of a negative test result given that the woman is not pregnant = 3/20 = 0.15

Using Bayes' theorem:
P(pregnant | negative test result) = (P(negative test result | pregnant) * P(pregnant)) / P(negative test result)
P(negative test result) = P(negative test result | pregnant) * P(pregnant) + P(negative test result | not pregnant) * P(not pregnant)

Calculating the values:
P(negative test result) = (0.0625 * 0.8) + (0.15 * 0.2) = 0.0875 + 0.03 = 0.1175

P(pregnant | negative test result) = (0.0625 * 0.8) / 0.1175 = 0.05 / 0.1175 ≈ 0.4255

Therefore, the probability of a woman being pregnant if the test indicates a negative result is approximately 0.4255 or 42.55%. If you are a physician and you have a patient who tested negative, you would advise further testing or monitoring, as there is still a significant chance of them being pregnant despite the negative result.

b) To find the probability of a false positive, which is the probability of getting a positive result if the woman is not actually pregnant, we can use the same approach, substituting the values into Bayes' theorem.

Let's denote:
P(positive test result | pregnant) = probability of a positive test result given that the woman is pregnant = 5/80 = 0.0625
P(positive test result | not pregnant) = probability of a positive test result given that the woman is not pregnant = 11/20 = 0.55

Using Bayes' theorem:
P(not pregnant | positive test result) = (P(positive test result | not pregnant) * P(not pregnant)) / P(positive test result)

Calculating the values:
P(positive test result) = P(positive test result | pregnant) * P(pregnant) + P(positive test result | not pregnant) * P(not pregnant)
P(positive test result) = (0.0625 * 0.8) + (0.55 * 0.2) = 0.05 + 0.11 = 0.16

P(not pregnant | positive test result) = (0.55 * 0.2) / 0.16 = 0.11 / 0.16 ≈ 0.6875

Therefore, the probability of a false positive, or getting a positive result if the woman is not actually pregnant, is approximately 0.6875 or 68.75%. If you are a physician and you have a patient who tested positive, you would advise further confirmation tests or examinations to rule out the possibility of a false positive.

c) Let's calculate the values for the given events:

i. P(pregnant | positive test result) = (P(positive test result | pregnant) * P(pregnant)) / P(positive test result)
= (0.0625 * 0.8) / 0.16 = 0.05

ii. P(positive test result | pregnant) = probability of a positive test result given that the woman is pregnant = 5/80 = 0.0625

The difference between the two events is the perspective in which they are considered. In event i, we are given that the test result is positive and we want to find the probability of the woman being pregnant. In event ii, we are given that the woman is pregnant and we want to find the probability of the test result being positive. Event i considers the test result as evidence of pregnancy, whereas event ii considers pregnancy as the condition for a positive test result.

The concept confusion of the inverse in this context arises because the probability of a positive test result given that the woman is pregnant (ii) is not the same as the probability of being pregnant given a positive test result (i). The confusion lies in thinking that these two probabilities should be equal, but they are not. It highlights the importance of understanding conditional probabilities and not assuming the inverse relationship holds true in all cases.