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Chapter 12: Problem 2

The rate of prostate cancer among men in 2002 was approximately 26 cases per 100000 people. Diagnosing this type of cancer saves the lives of about \(70 \%\) of those treated. At a hospital, the probability of diagnosing a person with this cancer correctly is \(78 \%\) and the probability of diagnosing a person without this cancer as having the disease is \(6 \%\). a) What is the probability of diagnosing a person as having this cancer? b) What percentage of those diagnosed with this cancer actually have it?

Short Answer

Expert verified
a) 6.01872%; b) 0.337%

Step by step solution

01

Calculate the Probability of a Man Having Prostate Cancer

The rate of prostate cancer per 100,000 people is 26 cases. This can be converted to a probability for an individual man: \[ P(C) = \frac{26}{100000} = 0.00026 \]
02

Calculate Probability of Correct Diagnosis Given Cancer

Given that a person has prostate cancer, the probability they are correctly diagnosed is 78%. Therefore,\[ P(D|C) = 0.78 \] where D represents a positive diagnosis.
03

Calculate the Probability of a False Positive Diagnosis

The probability that a person without the cancer is wrongly diagnosed as having it is 6%. Therefore,\[ P(D| eg C) = 0.06 \] where \(eg C\) means the person doesn't have cancer.
04

Calculate Probability of Not Having Prostate Cancer

The probability of a man not having prostate cancer is the complement of having it:\[ P(eg C) = 1 - P(C) = 1 - 0.00026 = 0.99974 \]
05

Calculate Total Probability of Diagnosing a Person as Having Cancer

The probability of diagnosing a person as having prostate cancer, \(P(D)\), is:\[ P(D) = P(D|C) \cdot P(C) + P(D|eg C) \cdot P(eg C) \]Substitute the known values:\[ P(D) = (0.78)(0.00026) + (0.06)(0.99974) \]\[ P(D) = 0.0002028 + 0.0599844 = 0.0601872 \]
06

Calculate the Probability of Actually Having Cancer Given a Diagnosis

Using Bayes' Theorem, the probability a diagnosis is correct, \(P(C|D)\), is:\[ P(C|D) = \frac{P(D|C) \cdot P(C)}{P(D)} \]Substitute the known values:\[ P(C|D) = \frac{0.78 \cdot 0.00026}{0.0601872} \]\[ P(C|D) \approx \frac{0.0002028}{0.0601872} \approx 0.00337 \text{ or } 0.337\% \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability
Probability is a measure of the likelihood of a particular event occurring. Imagine flipping a coin, where each flip has a 50% chance of landing on heads and a 50% chance of landing on tails. In our cancer diagnosis example, probability helps us understand the chances of various outcomes, such as having prostate cancer or not.

Let's break it down:
  • The probability of an event is calculated by dividing the number of successful outcomes by the total number of possible outcomes.
  • In this example, the probability of a man having prostate cancer, denoted as \( P(C) \), is \( 0.00026 \), meaning there is a very small chance a random man will have this cancer.
  • The complement of this probability gives us \( P(eg C) = 0.99974 \), indicating the probability of not having prostate cancer.
Calculating probabilities helps provide a clearer understanding and form the basis for deeper analyses, like diagnosing diseases accurately.
False Positive
A false positive occurs when a test incorrectly indicates the presence of a condition that is not actually present. In medical terms, it's like sounding a false alarm. This can cause unnecessary stress and further testing for the person involved.

In our example:
  • The probability of a false positive diagnosis, denoted as \( P(D| eg C) \), is 6%.
  • This means that out of men who do not have prostate cancer, 6% will still receive a diagnosis that they do have it.
Managing and understanding false positives is critical because they can lead to unnecessary treatments and anxiety. This is why accuracy in medical tests is crucial, to avoid both false positives and false negatives, where a condition is present but not detected.
Conditional Probability
Conditional probability is the likelihood of an event occurring given that another event has already occurred. It lets us refine our understanding of probabilities based on additional information.

Using our example, if a man receives a diagnosis, conditional probability helps us determine how likely it is that he truly has prostate cancer.

This is calculated using Bayes' Theorem:
  • The theorem combines different probability measurements to give us \( P(C|D) \), or the probability of having cancer given a diagnosis.
  • In our scenario, the calculation results in about 0.337%, showing that despite receiving a diagnosis, there鈥檚 still a low chance of having prostate cancer.
Bayes' Theorem is powerful in many fields, not just medicine, to make more informed predictions based on specific conditions or events.

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Most popular questions from this chapter

Five cards are chosen at random from a deck of 52 cards. Find the probability that the set contains a) 3 kings b) 4 hearts and 1 diamond.

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