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Chapter 13: Problem 33

Tendon surgery, continued. You have torn a tendon and are facing surgery to repair it. The surgeon explains the risks to you: infection occurs in \(3 \%\) of such operations, the repair fails in \(14 \%\), and both infection and failure occur together in \(1 \%\). What is the probability of infection, given that the repair is successful? Follow the four-step process in your answer.

Short Answer

Expert verified
The probability of infection given a successful repair is approximately 2.33%.

Step by step solution

01

Understand the Problem

The problem involves calculating a conditional probability, specifically the probability of infection given that the repair was successful. We know the overall probability of infection, failure, and both occurring together.
02

Identify Known Probabilities

Given data includes: \( P(I) = 0.03 \) (probability of infection), \( P(F) = 0.14 \) (probability of repair failure), and \( P(I \cap F) = 0.01 \) (probability of both infection and failure together). We need to find \( P(I|F^c) \), where \( F^c \) is the event that the repair is successful.
03

Determine Probability of Successful Repair

Successful repair occurs when there is no failure, i.e., \( F^c \). Calculate \( P(F^c) \) using the complement rule: \( P(F^c) = 1 - P(F) = 1 - 0.14 = 0.86 \).
04

Calculate Infection Given Successful Repair

Use the formula for conditional probability: \( P(I|F^c) = \frac{P(I \cap F^c)}{P(F^c)} \). First calculate \( P(I \cap F^c) \) using \( P(I \cap F^c) = P(I) - P(I \cap F) = 0.03 - 0.01 = 0.02 \). Then, calculate \( P(I|F^c) = \frac{0.02}{0.86} \approx 0.0233 \).
05

Verify and Interpret Results

Check calculations for any errors and interpret the results: \( P(I|F^c) \approx 0.0233 \) means there is approximately a 2.33% probability of developing an infection if the tendon repair is successful.

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Key Concepts

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

Probability Calculation
Probability is the cornerstone of statistics and is used to measure the likelihood that a certain event will happen. Every probability value is a number between 0 and 1, where 0 means the event will not happen, and 1 means it will surely happen.
To compute the probability of an event, we often rely on historical data or theoretical models.
In our exercise, we calculate different types of probabilities such as the basic probability of events like infection and failure, and their combination. Calculating probability helps to anticipate outcomes and make informed decisions.
Complement Rule
The complement rule is a useful tool in probability that simplifies the calculation of probabilities by using the known probabilities of the opposite event.
Specifically, the probability of an event not happening is 1 minus the probability of it happening, expressed as
  • \( P(A^c) = 1 - P(A) \)
where \( A^c \) is the complement of \( A \).
In this exercise, we calculate the probability of a tendon surgery repair being successful using the complement rule. This helps when determining the likelihood of a successful outcome () when knowing its opposite counterpart (failure).
Using complements often makes probability calculations easier and provides a quick way to verify results.
Probability of Infection
In medical contexts, understanding the probability of infection is crucial for both risk assessment and patient care.
Here, the probability of infection during surgery, \( P(I) \), is given as 0.03 or 3%, indicating infection happens in about one out of every 33 surgeries.
This calculation becomes more complex when we need to compute it under different conditions such as when surgery is successful. To find this probability, knowing the probability of both infection and failure together is essential. Understanding these probabilities can assist in making medical and personal decisions about whether to proceed with surgery.
Probability of Failure
The probability of failure, noted as \( P(F) \), often represents the inherent risk tied to any procedure or system.
In the context of surgery, failure could signify that the tendon repair does not hold. Given as 14% in our exercise, this represents significant odds where additional considerations for risk mitigation might be advised.
Understanding the probability of failure guides medical professionals and patients in evaluating the potential costs and benefits of the procedure. More broadly, recognizing these risks can prompt actions to manage or reduce them, ensuring every effort is made to increase the chance of successful outcomes.

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Most popular questions from this chapter

Universal blood donors. People with type O-negative blood are referred to as universal donors, although if you give type O-negative blood to any patient, you run the risk of a transfusion reaction due to certain antibodies present in the blood. However, any patient can receive a transfusion of O-negative red blood cells. Only \(7.2 \%\) of the American population have O-negative blood. If 10 people appear at random to give blood, what is the probability that at least one of them is a universal donor?

Lactose intolerance. Lactose intolerance causes difficulty digesting dairy products that contain lactose (milk sugar). It is particularly common among people of African and Asian ancestry. In the United States (ignoring other groups and people who consider themselves to belong to more than one race), \(82 \%\) of the population is white, \(14 \%\) is black, and \(4 \%\) is Asian. Moreover, \(15 \%\) of whites, \(70 \%\) of blacks, and \(90 \%\) of Asians are lactose intolerant. 22 (a) What percent of the entire population is lactose intolerant? (b) What percent of people who are lactose intolerant are Asian?

Independent? In 2015 , the Report on the LC Berkeley Faculty Salary Equity Study shows that 87 of the university's 222 assistant professors were women, along with 137 of the 324 associate professors and 243 of the 972 full professors. (a) What is the probability that a randomly chosen Berkeley professor (of any rank) is a woman? (b) What is the conditional probability that a randomly chosen professor is a woman, given that the person chosen is a full professor? (c) Are the rank and sex of Berkeley professors independent? How do you know?

(Optional topic) College degrees and sex. According to the National Center for Educational Statistics, in \(201426 \%\) of college degrees were associate degrees, \(49 \%\) were bachelor's, \(20 \%\) were master's, and \(5 \%\) were doctor's (including professional degrees such as MD, DDS, and law degrees). Women earned \(61 \%\) of the associate degrees, \(57 \%\) of the bachelor's, \(60 \%\) of the master's, and \(52 \%\) of the doctor's. (a) What are the prior probabilities for the four types of degrees? (b) Find the posterior probabilities for the type of degree given that the recipient is female. Is the relationship between the prior and posterior probabilities what you would expect? Explain briefly.

Of people who died in the United States in recent years, \(86 \%\) were white, \(12 \%\) were black, and \(2 \%\) were Asian. (This ignores a small number of deaths among other races.) Diabetes caused \(2.8 \%\) of deaths among whites, \(4.4 \%\) among blacks, and \(3.5 \%\) among Asians. The probability that a randomly chosen death is a white person who died of diabetes is about (a) \(0.107\). (b) \(0.030\). (c) \(0.024\).
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