Question

If a woman takes an early pregnancy test, she will either test positive, meaning that the test says she is pregnant, or test negative, meaning that the test says she is not pregnant. Suppose that if a woman really is pregnant, there is a \(98 \%\) chance that she will test positive. Also, suppose that if a woman really is not pregnant, there is a \(99 \%\) chance that she will test negative. (a) Suppose that 1,000 women take early pregnancy tests and that 100 of them really are pregnant. What is the probability that a randomly chosen woman from this group will test positive? (b) Suppose that 1,000 women take early pregnancy tests and that 50 of them really are pregnant. What is the probability that a randomly chosen woman from this group will test positive?

Short answer

The probability that a randomly chosen woman from the first group will test positive is 10.7%. The probability that a randomly chosen woman from the second group will test positive is 5.85%.

Step-by-step solution

Determine The Number of Women Who Test Positive and Are Pregnant

Given that 100 out of 1,000 women really are pregnant and the test has a 98% success rate, we calculate the number of pregnant women who test positive: 0.98 * 100 = 98 women.

Determine The Number of Women Who Test Positive but Are Not Pregnant

The remaining women (1,000 - 100 = 900) are not pregnant. Since there is a 1% chance of a false positive, we calculate the number of non-pregnant women who will test positive: 0.01 * 900 = 9 women.

Calculate The Total Number of Positive Tests

To find the total number of positive tests, we add the number of pregnant women who test positive to the number of non-pregnant women who test positive: 98 (from Step 1) + 9 (from Step 2) = 107.

Calculate The Probability of a Positive Test Result

The probability of a positive test result is the total number of positive tests divided by the total number of tests: 107 / 1,000 = 0.107 or 10.7%.

Calculate The Adjusted Number of Pregnant Women

Now, considering that 50 out of 1,000 women really are pregnant, we calculate the number of pregnant women who test positive with the same 98% success rate: 0.98 * 50 = 49 women.

Calculate The Adjusted Number of Non-Pregnant Women Who Test Positive

For the remaining women (1,000 - 50 = 950) who are not pregnant, we calculate the number of false positives: 0.01 * 950 = 9.5 women.

Calculate The Adjusted Total Number of Positive Tests

We now add the adjusted numbers of true positives and false positives: 49 (from Step 5) + 9.5 (from Step 6) = 58.5.

Calculate The Adjusted Probability of a Positive Test Result

The adjusted probability of a positive test result is: 58.5 / 1,000 = 0.0585 or 5.85%.

Key concepts

Conditional Probability

Understanding conditional probability is essential in various fields, particularly in the life sciences, where it's used to assess the likelihood of certain outcomes given specific conditions.

Conditional probability is expressed as P(A|B), which is the probability of event A occurring given that event B has occurred. This concept is fundamental when dealing with diagnostic tests like early pregnancy tests, as in the exercise provided. For instance, the probability that a woman who is actually pregnant gets a positive result (event A) given that she takes a pregnancy test (event B) can be considered as conditional probability.

To illuminate this with an example from the exercise, 100 out of 1,000 women are truly pregnant (event B). The probability these women will test positive (event A) is 98%, which means almost all pregnant women will correctly get a positive result. On the other hand, the probability that a woman who is not pregnant gets a positive result is also based on the premise that she has taken the test.

False Positive Rate

The false positive rate is a critical measure in medical testing and relates to the reliability of a test result. It refers to the proportion of non-pregnant individuals who incorrectly receive a positive test result. A lower false positive rate indicates a more reliable diagnostic test.

In the context of the early pregnancy test example, the test has a 99% chance of correctly indicating non-pregnancy. This implies a 1% false positive rate because there's a 1% chance the test will indicate pregnancy in a woman who is not actually pregnant. The calculation is straightforward: take the number of non-pregnant women, 900 (1,000 - 100 pregnant women), and then apply the false positive rate of 1% to find out how many non-pregnant women would incorrectly test positive, which in this case is 9 women.

In practical terms, it's important to have a low false positive rate to minimize the emotional distress and medical costs associated with unnecessary follow-up procedures.

Predictive Value of Tests

The predictive value of tests is a term used to describe how well a test can predict the presence (positive predictive value) or absence (negative predictive value) of a condition. It considers the number of true positive results relative to all positive results (both true positives and false positives).

In the exercise, the total number of positive tests (the sum of true positives and false positives) is 107. To find the predictive value of the test, one must consider how many of these positives truly reflect the condition being tested for, which in this case is pregnancy. With 98 true positive tests and 9 false positive tests, the probability that a positive test correctly reflects pregnancy can be calculated.

This concept is extremely important in clinical settings because it helps patients and healthcare providers understand the likelihood that a positive (or negative) test result is accurate, which informs subsequent medical decisions. A high predictive value of a test means the test results are reliable indicators of the condition.