91Ó°ÊÓ

A study of the impact of seeking a second opinion about a medical condition is described in the paper "Evaluation of Outcomes from a National Patient- Initiated Second-Opinion Program". Based on a review of 6791 patient-initiated second opinions, the paper states the following: "Second opinions often resulted in changes in diagnosis (14.8\%), treatment \((37.4 \%),\) or changes in both \((10.6 \%)\)." Consider the following two events: \(D=\) event that second opinion results in a change in diagnosis \(T=\) event that second opinion results in a change in treatment a. What are the values of \(P(D), P(T),\) and \(P(D \cap T) ?\) b. Use the given probability information to set up a hypothetical 1000 table with columns corresponding to \(D\) and \(D^{C}\) and rows corresponding to \(T\) and \(T^{C}\). c. What is the probability that a second opinion results in neither a change in diagnosis nor a change in treatment? d. What is the probability that a second opinion results is a change in diagnosis or a change in treatment?

Step by step solution

01

Determine the probability of each event.

P(D) = 14.8% = 0.148 P(T) = 37.4% = 0.374 P(D ∩ T) = 10.6% = 0.106 #Step 2: Create the 1000 Table#

02

Set up a hypothetical 1000 table with columns corresponding to D and D^C and rows corresponding to T and T^C.

We have a table like this: D | D^C ----------------|--------- T | 106 | 268 ----------------|--------- T^C | 42 | 584 Here, the values in the table are calculated as follows: - Total cases = 1000 - D and T: 10.6% of 1000 = 106 - D and T^C: 14.8% - 10.6% = 4.2% of 1000 = 42 - D^C and T: 37.4% - 10.6% = 26.8% of 1000 = 268 - D^C and T^C: 1000 - (106 + 42 + 268) = 584 #Step 3: Probability of Neither Change in Diagnosis Nor Change in Treatment#

03

Determine the probability that a second opinion results in neither a change in diagnosis nor a change in treatment.

Looking at the 1000 table, we find that there are 584 cases where neither D nor T occur. Therefore, the probability is: \(P(D^C \cap T^C) = \frac{584}{1000} = 0.584\) #Step 4: Probability of Change in Diagnosis or Change in Treatment#

04

Determine the probability that a second opinion results in a change in diagnosis or a change in treatment.

Using the formula for the probability of the union of two events: \(P(D \cup T) = P(D) + P(T) - P(D \cap T)\) Then we get: \(P(D \cup T) = 0.148 + 0.374 - 0.106 = 0.416\) So, the probability that a second opinion results in a change in diagnosis or a change in treatment is 41.6%.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Conditional Probability

Conditional probability is an important concept in probability theory. It helps us understand how the probability of one event changes when we know another event has occurred. In simple terms, it answers the question: "Given that event A has happened, what is the probability that event B will happen too?"
The formula for conditional probability is given by \( P(B|A) = \frac{P(B \cap A)}{P(A)} \), where:

• \( P(B|A) \) is the probability of event B happening given that event A has happened.
• \( P(B \cap A) \) is the probability that both events A and B occur together.
• \( P(A) \) is the probability that event A occurs.

Conditional probability is particularly useful in making predictions and informed decisions under uncertainty. For instance, in the context of the exercise, if we were to ask about the likelihood of a change in treatment given there was a change in diagnosis (or vice versa), we'd use conditional probability. It tells us how one factor statistically influences or changes another in the real world.

Joint Probability

Joint probability involves the likelihood of two or more events occurring simultaneously. This is calculated as \( P(A \cap B) \), which represents the probability that both event A and event B happen. Joint probability is a foundational concept when analyzing scenarios where multiple events might affect each other.
Think of it as a way of assessing the interdependence between two events. In our exercise, we calculate the joint probability of a second opinion resulting in a change in both diagnosis and treatment, which is given by \( P(D \cap T) = 0.106 \). This means there is a 10.6% chance that both changes occur simultaneously.
When working with joint probabilities, its usefulness is evident when you're looking to understand the relationship and co-occurrence of different events. It's a necessary component for further calculations, such as determining conditional probabilities and the probability of the union of events.

Probability of Union

The probability of the union of two events refers to the likelihood that at least one of the events occurs. This is represented as \( P(A \cup B) \). It can be thought of as a combination of the probabilities of individual events, without double-counting the events that occur simultaneously.
The formula for calculating this is given by:

• \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \)

In simpler terms, it adds up the probability of each event and subtracts the probability of both events occurring together, as this would otherwise be counted twice. In our specific exercise, the probability that a second opinion results in either a change in diagnosis or a change in treatment is 41.6%.
This measure is especially useful when considering scenarios where multiple outcomes might satisfy a success condition, making it essential for broader probability calculations in many real-world applications.