Question

Different illnesses can produce identical symptoms. Suppose a particular set of symptoms, which we will denote as event \(H,\) occurs only when any one of three illnesses- \(A, B,\) or \(C-\) occurs. (For the sake of simplicity, we will assume that illnesses \(A, B,\) and \(C\) are mutually exclusive.) Studies show these probabilities of getting the three illnesses: $$ \begin{array}{l} P(A)=.01 \\ P(B)=.005 \\ P(C)=.02 \end{array} $$ The probabilities of developing the symptoms \(H,\) given a specific illness, are $$ \begin{array}{l} P(H \mid A)=.90 \\ P(H \mid B)=.95 \\ P(H \mid C)=.75 \end{array} $$ Assuming that an ill person shows the symptoms \(H\), what is the probability that the person has illness \(A ?\)

Short answer

Answer: The probability that a person has illness A, given that they show the symptoms H, is approximately 0.3133 or 31.33%.

Step-by-step solution

Understand the problem and given information

First, we need to understand the given information. We have the probabilities of developing illnesses A, B, and C, and the probabilities of showing symptoms H given each specific illness. Our goal is to find the probability that a person has illness A, given that they show the symptoms H.

Write down the conditional probability formula

We need to find the probability of having illness A, given the symptoms H. In terms of conditional probability, we are looking for P(A|H). We can use the conditional probability formula to help us find this probability: P(A|H) = P(H|A) * P(A) / P(H)

Apply Bayes' theorem to find P(H)

To find the probability P(H), we can use Bayes' theorem. Since H is the set of symptoms that can occur only when any one of the three illnesses A, B, or C occurs, the probability of H can be written as: P(H) = P(H∩A) + P(H∩B) + P(H∩C) Now, we can substitute the probabilities of developing symptoms H given the specific illnesses: P(H) = P(H|A) * P(A) + P(H|B) * P(B) + P(H|C) * P(C)

Substitute the known values and calculate P(H)

Now, we have all the probabilities we need to calculate P(H). We substitute the given values of P(A), P(B), P(C), P(H|A), P(H|B), and P(H|C) into the equation: P(H) = (0.90 * 0.01) + (0.95 * 0.005) + (0.75 * 0.02) P(H) = 0.009 + 0.00475 + 0.015 P(H) = 0.02875

Calculate P(A|H) using the conditional probability formula

Now, we have the value of P(H) that we need to calculate P(A|H). We use the conditional probability formula with the given values of P(H|A) and P(A), and the calculated value of P(H): P(A|H) = (0.90 * 0.01) / 0.02875 P(A|H) = 0.009 / 0.02875 P(A|H) ≈ 0.3133 Thus, the probability that a person has illness A, given that they show the symptoms H, is approximately 0.3133 or 31.33%.

Key concepts

Bayes' Theorem

Bayes' theorem is a fundamental formula used to determine the probability of an event based on prior knowledge of conditions related to the event. It is especially useful when dealing with problems of diagnostic testing or whenever related probabilities are updated with new information.

The theorem is expressed mathematically as:
\[ P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)} \
where:\

• \( P(A|B) \) is the probability of event \( A \) given that \( B \) has occurred.
• \( P(B|A) \) is the probability of event \( B \) given that \( A \) has occurred.
• \( P(A) \) and \( P(B) \) are the probabilities of events \( A \) and \( B \) independently occurring.

In our exercise, Bayes' theorem helps us to calculate the likelihood that a person has a specific illness \( A \) based on observing symptom \( H \). This is essential for medical diagnosis, where symptoms can be caused by various diseases, and we need to pinpoint the correct one based on the evidence.

Mutually Exclusive Events

Mutually exclusive events are events that cannot occur at the same time. In the language of probability theory, if two events are mutually exclusive, the probability of both occurring simultaneously is zero.

Mathematically, for any two mutually exclusive events \( A \) and \( B \), we have:\[ P(A \cap B) = 0 \]
This property simplifies the calculation of combined probabilities. Since illnesses \( A, B, \) and \( C \) in the given exercise are mutually exclusive, it is impossible for a person to have more than one of these illnesses at the same time. When calculating the overall probability of symptom \( H \), we can simply sum the probabilities of \( H \) occurring with each illness without worrying about overlap.

Probability Theory

Probability theory is a branch of mathematics concerned with analyzing random phenomena. It provides the framework to model uncertainty and quantify the likelihood of potential outcomes. Basic concepts include independent events, conditional probability, and mutually exclusive events, which we've discussed.

Every event has a probability between 0 and 1, with 0 indicating impossibility and 1 signifying certainty. The sum of probabilities of all mutually exclusive possible outcomes of a random experiment is always equal to 1.

To effectively apply probability theory, such as in our textbook problem, one must correctly identify the relevant events, understand their interrelationships, and use the right formulas and principles to calculate probabilities. The correct application of probability theory is crucial in various fields, from healthcare to finance, helping to make informed decisions under uncertainty.