Problem 49 - A First Course in Probability

Question

Prostate cancer is the most common type of cancer found in males. As an indicator of whether a male has prostate cancer, doctors often perform a test that measures the level of the prostate specific antigen (PSA) that is produced only by the prostate gland. Although PSA levels are indicative of cancer, the test is notoriously unreliable. Indeed, the probability that a noncancerous man will have an elevated PSA level is approximately. \(135,\) increasing to approximately .268 if the man does have cancer. If, on the basis of other factors, a physician is 70 percent certain that a male has prostate cancer, what is the conditional probability that he has the cancer given that (a) the test indicated an elevated PSA level? (b) the test did not indicate an elevated PSA level? Repeat the preceding calculation, this time assuming that the physician initially believes that there is a 30 percent chance that the man has prostate cancer.

Step-by-Step Solution

Answer

In conclusion, the conditional probabilities of having prostate cancer given the test results are: - If the physician is 70% certain of cancer, P(C|E) ≈ 0.848 and P(C|¬E) ≈ 0.659. - If the physician is 30% certain of cancer, P(C|E) ≈ 0.475 and P(C|¬E) ≈ 0.264.

Step 1: Calculate P(E) and P(¬E)

We will first find P(E) and P(¬E) using the law of total probability: P(E) = P(E|C)P(C) + P(E|N)P(N) P(E) = 0.268(0.7) + 0.135(0.3) P(E) ≈ 0.2217 Similarly, we calculate P(¬E): P(¬E) = 1 - P(E) ≈ 1 - 0.2217 ≈ 0.7783

Step 2: Apply Bayes' theorem to find P(C|E) and P(C|¬E)

Now we can use Bayes' theorem to find the conditional probabilities: (a) P(C|E) = P(E|C)P(C) / P(E) P(C|E) = 0.268(0.7) / 0.2217 P(C|E) ≈ 0.848 (b) P(C|¬E) = P(¬E|C)P(C) / P(¬E) We first need to find P(¬E|C), which is equal to 1 - P(E|C) = 1 - 0.268 = 0.732 P(C|¬E) = 0.732(0.7) / 0.7783 P(C|¬E) ≈ 0.659 Let's now assume that the physician initially believes that there is a 30% chance that the man has prostate cancer. Therefore, we'll have P(C) = 0.3 and P(N) = 0.7.

Step 3: Find P(E) and P(¬E) with updated probabilities

In this case, we need to recalculate P(E) and P(¬E): P(E) = P(E|C)P(C) + P(E|N)P(N) P(E) = 0.268(0.3) + 0.135(0.7) P(E) ≈ 0.1694 P(¬E) = 1 - P(E) ≈ 1 - 0.1694 ≈ 0.8306

Step 4: Apply Bayes' theorem to find the new P(C|E) and P(C|¬E)

We now apply Bayes' theorem again to find the new conditional probabilities: (a) P(C|E) = P(E|C)P(C) / P(E) P(C|E) = 0.268(0.3) / 0.1694 P(C|E) ≈ 0.475 (b) P(C|¬E) = P(¬E|C)P(C) / P(¬E) P(C|¬E) = 0.732(0.3) / 0.8306 P(C|¬E) ≈ 0.264 In conclusion, the conditional probabilities of having prostate cancer given the test results are: - P(C|E) ≈ 0.848 if the physician is 70% certain of cancer, and P(C|E) ≈ 0.475 if they are 30% certain. - P(C|¬E) ≈ 0.659 if the physician is 70% certain of cancer, and P(C|¬E) ≈ 0.264 if they are 30% certain.

Key Concepts

Conditional Probability

Conditional probability is a fundamental concept in probability theory that helps us understand how the probability of an event changes when we have extra information. In simple terms, conditional probability asks: "Given that one event has happened, what is the probability another event will happen?" It's denoted as \(P(A|B)\), which reads as "the probability of event \(A\) given event \(B\)".

In the context of the exercise, we are interested in finding the probability that a male actually has prostate cancer given that a PSA test result is either elevated or not. Here, \(C\) represents having cancer, and \(E\) represents an elevated test result. So, \(P(C|E)\) is the probability of having cancer given an elevated PSA. Moreover, \(P(C|¬E)\) represents the probability of having cancer given a non-elevated PSA.

• If a physician is 70% certain a patient has cancer and the PSA is elevated, \(P(C|E)\) is calculated to be 0.848.
• When the PSA is not elevated, \(P(C|¬E)\) is 0.659.
• If the physician's initial certainty changes to 30%, these probabilities adjust to 0.475 and 0.264, respectively.

Law of Total Probability

The law of total probability helps us calculate the overall probability of an event by considering all possible ways that event can occur. It's particularly useful when we need to break down complex probability problems into simpler parts.

In the exercise, the law of total probability is used to compute \(P(E)\), the probability of having an elevated PSA result, and \(P(eg E)\), the probability of not having an elevated result. This involves considering all the scenarios: having cancer and not having cancer. The formula for \(P(E)\) is:

\[ P(E) = P(E|C)P(C) + P(E|N)P(N) \]

Here, \(P(E|C)\) is the probability of an elevated PSA given cancer, while \(P(E|N)\) is the probability of an elevated PSA without cancer. \(P(C)\) and \(P(N)\) are the probabilities of having cancer and not having cancer, respectively.

Using this approach:

• With initial cancer certainty at 70%, \(P(E) \approx 0.2217\) and \(P(eg E) \approx 0.7783\).
• With 30% initial certainty, \(P(E)\) becomes approximately 0.1694, and \(P(eg E)\) is approximately 0.8306.

Probability Calculations

Probability calculations involve determining the likelihood of different outcomes using known probabilities. In this problem, these calculations are done by employing the law of total probability and Bayes' theorem.

The steps for performing these calculations are straightforward but require methodical application of these tools:

• Firstly, calculate \(P(E)\) and \(P(eg E)\) using the law of total probability to understand the overall scenario.
• Next, use Bayes' theorem to find conditional probabilities like \(P(C|E)\) and \(P(C|eg E)\). This involves determining how the likelihood of cancer changes with PSA results.
• Finally, update your calculations if initial beliefs change, which impacts all derived probabilities.

By following these steps, you can accurately evaluate and interpret the probabilities that relate to whether a male has prostate cancer, given different situations and evidence such as PSA test results.