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Chapter 4: Problem 49

A study of 31,000 hospital admissions in New York State found that \(4 \%\) of the admissions led to treatment-caused injuries. One-seventh of these treatment-caused injuries resulted in death, and one-fourth were caused by negligence. Malpractice claims were filed in one out of 7.5 cases involving negligence, and payments were made in one out of every two claims. a. What is the probability a person admitted to the hospital will suffer a treatment-caused injury due to negligence? b. What is the probability a person admitted to the hospital will die from a treatmentcaused injury? c. In the case of a negligent treatment-caused injury, what is the probability a malpractice claim will be paid?

Short Answer

Expert verified
a. 0.01, b. 0.0057, c. 0.0645

Step by step solution

01

Calculate Treatment-Caused Injuries

To find the number of treatment-caused injuries, calculate 4% of 31,000 admissions. \[ \text{Injuries} = 0.04 \times 31000 = 1240 \]
02

Calculate Injuries Due to Negligence

Next, find one-fourth of these injuries caused by negligence. \[ \text{Negligent Injuries} = \frac{1}{4} \times 1240 = 310 \]
03

Probability of Negligent Injury

The probability that a person admitted will suffer a treatment-caused injury due to negligence is the ratio of negligent injuries to total admissions.\[ P(\text{Negligent Injury}) = \frac{310}{31000} = 0.01 \]
04

Calculate Injuries Resulting in Death

Calculate one-seventh of the 1240 injuries that resulted in death.\[ \text{Injuries Resulting in Death} = \frac{1}{7} \times 1240 = 177.14 \approx 177 \]
05

Probability of Death From Injury

The probability that a person will die from a treatment-caused injury is the ratio of deaths to total admissions.\[ P(\text{Death From Injury}) = \frac{177}{31000} \approx 0.0057 \]
06

Calculate Claim Filed for Negligent Injury

Calculate claims filed by finding one out of every 7.5 negligent cases.\[ \text{Claims Filed} = \frac{310}{7.5} \approx 41.33 \approx 41 \]
07

Calculate Claims Paid

Calculate the number of claims paid by finding half of the claims filed.\[ \text{Claims Paid} = \frac{41}{2} = 20.5 \approx 20 \]
08

Probability of a Claim Being Paid

The probability that a malpractice claim will be paid given negligence is the ratio of claims paid to the negligent injuries.\[ P(\text{Claim Paid | Negligent Injury}) = \frac{20}{310} \approx 0.0645 \]

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Key Concepts

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

Treatment-Caused Injuries
When people are admitted to hospitals, we may assume that they are seeking help to recover from their ailments. However, there is always a risk that treatment itself could cause harm. These are known as treatment-caused injuries. In the context of a study in New York State, it's been found that 4% of hospital admissions resulted in treatment-caused injuries.
This means that out of every 100 patients, 4 might experience an injury directly related to the treatment they are receiving. This can include complications from surgeries, drug reactions, or errors in medical procedures. Understanding and calculating the probability of treatment-caused injuries is crucial in assessing hospital performance and patient safety. It involves multiplying the percentage (as a decimal) by the total number of admissions to get the total number of such injuries.
In this case, it involved calculating 4% of 31,000 admissions which resulted in 1,240 treatment-caused injuries.
Negligence in Healthcare
Negligence in healthcare refers to actions, or lack thereof, by healthcare professionals that fall below the accepted standard, leading to harm. Not every treatment-caused injury is due to negligence, but when it is, it indicates that the harm could potentially have been avoided if due care were taken.
For the study, one-fourth of the treatment-caused injuries were attributed to negligence. This means out of 1,240 treatment-caused injuries, 310 were due to negligence. To find this, you multiply the number of treatment-caused injuries by one-fourth.
From a probability standpoint, if you think of it in terms of hospital admissions, those who might suffer from negligent treatment-caused injuries amount to 310 out of the 31,000 admissions. Dividing these gives us the probability or chance of a person experiencing a negligent injury, which was found to be 0.01 or 1%.
Malpractice Claims
When a patient suffers harm due to negligence, they might file a malpractice claim against the healthcare provider seeking compensation. However, not every negligent injury results in a malpractice claim. In the exercise, it was observed that only one out of every 7.5 cases of negligence led to a claim being filed.
By dividing the total negligent injuries (310) by 7.5, we find approximately 41 such claims were filed out of the negligent events. Further yet, not every claim results in a payment. In our example, only one out of every two claims led to a settlement.
To assess the likelihood of a malpractice claim being paid, we evaluate how many of the filed claims were successful. Calculating half of the claims filed (41), we approximate that 20 claims were successfully settled. The probability, therefore, of a claim being paid where there was negligence is calculated by dividing these successful claims by the number of negligent injuries, resulting in approximately 0.0645 or 6.45%.
Hospital Admissions Statistics
Statistics about hospital admissions give us critical insights into healthcare delivery and patient outcomes. Such statistics can be used to benchmark hospital performance and analyze trends in patient safety. For instance, the New York study provided vital statistics, with 31,000 admissions analyzed to glean the rate of treatment-caused injuries, negligence, and malpractice claims.
By breaking down large data sets into these probabilities, hospitals can allocate resources towards better training, improved processes, and systems that advance patient care. Understanding what percentage of admissions led to adverse outcomes also helps in forming healthcare policies, aiming to reduce treatment-caused injuries and improve patient safety.
  • 31,000 admissions as the total basis for probability calculations
  • Insights on treatment-caused injury rates
  • Determination of injury causes from negligence or other factors
  • Potential for malpractice claims based on negligence probability
These statistics help in forming a holistic view of what happens during hospital admissions and are foundational for improvements in the healthcare system.

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Most popular questions from this chapter

In an article about investment growth, Money magazine reported that drug stocks show powerful long-term trends and offer investors unparalleled potential for strong and steady gains. The federal Health Care Financing Administration supports this conclusion through its forecast that annual prescription drug expenditures will reach \(\$ 366\) billion by 2010 , up from \(\$ 117\) billion in 2000 . Many individuals age 65 and older rely heavily on prescription drugs. For this group, \(82 \%\) take prescription drugs regularly, \(55 \%\) take three or more prescriptions regularly, and \(40 \%\) currently use five or more prescriptions. In contrast, \(49 \%\) of people under age 65 take prescriptions regularly, with \(37 \%\) taking three or more prescriptions regularly and \(28 \%\) using five or more prescriptions (Money, September 2001 ). The U.S. Census Bureau reports that of the 281,421,906 people in the United States, 34,991,753 are age 65 years and older (U.S. Census Bureau, Census 2000 ). a. Compute the probability that a person in the United States is age 65 or older. b. Compute the probability that a person takes prescription drugs regularly. c. Compute the probability that a person is age 65 or older and takes five or more prescriptions. d. Given a person uses five or more prescriptions, compute the probability that the person is age 65 or older.

A Morgan Stanley Consumer Research Survey sampled men and women and asked each whether they preferred to drink plain bottled water or a sports drink such as Gatorade or Propel Fitness water (The Atlanta Journal-Constitution, December 28,2005 ). Suppose 200 men and 200 women participated in the study, and 280 reported they preferred plain bottled water. Of the group preferring a sports drink, 80 were men and 40 were women. Let \(M=\) the event the consumer is a man \(W=\) the event the consumer is a woman \(B=\) the event the consumer preferred plain bottled water \(S=\) the event the consumer preferred sports drink a. What is the probability a person in the study preferred plain bottled water? b. What is the probability a person in the study preferred a sports drink? c. What are the conditional probabilities \(P(M | S)\) and \(P(W | S) ?\) d. What are the joint probabilities \(P(M \cap S)\) and \(P(W \cap S) ?\) e. Given a consumer is a man, what is the probability he will prefer a sports drink? f. Given a consumer is a woman, what is the probability she will prefer a sports drink? g. Is preference for a sports drink independent of whether the consumer is a man or a woman? Explain using probability information.

Suppose that we have a sample space \(S=\left\\{E_{1}, E_{2}, E_{3}, E_{4}, E_{5}, E_{6}, E_{7}\right\\},\) where \(E_{1}, E_{2}, \ldots,\) \(E_{7}\) denote the sample points. The following probability assignments apply: \(P\left(E_{1}\right)=.05\) \(P\left(E_{2}\right)=.20, P\left(E_{3}\right)=.20, P\left(E_{4}\right)=.25, P\left(E_{5}\right)=.15, P\left(E_{6}\right)=.10,\) and \(P\left(E_{7}\right)=.05 .\) Let $$\begin{array}{l} A=\left\\{E_{1}, E_{4}, E_{6}\right\\} \\ B=\left\\{E_{2}, E_{4}, E_{7}\right\\} \\ C=\left\\{E_{2}, E_{3}, E_{5}, E_{7}\right\\} \end{array}$$ a. Find \(P(A), P(B),\) and \(P(C)\) b. Find \(A \cup B\) and \(P(A \cup B)\) c. Find \(A \cap B\) and \(P(A \cap B)\) d. Are events \(A\) and \(C\) mutually exclusive? e. Find \(B^{c}\) and \(P\left(B^{c}\right)\)

The prior probabilities for events \(A_{1}\) and \(A_{2}\) are \(P\left(A_{1}\right)=.40\) and \(P\left(A_{2}\right)=.60 .\) It is also known that \(P\left(A_{1} \cap A_{2}\right)=0 .\) Suppose \(P\left(B \text { ? } A_{1}\right)=.20\) and \(P\left(B | A_{2}\right)=.05\) a. Are \(A_{1}\) and \(A_{2}\) mutually exclusive? Explain. b. Compute \(P\left(A_{1} \cap B\right)\) and \(P\left(A_{2} \cap B\right)\) c. Compute \(P(B)\) d. Apply Bayes' theorem to compute \(P\left(A_{1} | B\right)\) and \(P\left(A_{2} | B\right)\).

Consider the experiment of rolling a pair of dice. Suppose that we are interested in the sum of the face values showing on the dice. a. How many sample points are possible? (Hint: Use the counting rule for multiple-step experiments.) b. List the sample points. c. What is the probability of obtaining a value of \(7 ?\) d. What is the probability of obtaining a value of 9 or greater? e. Because each roll has six possible even values \((2,4,6,8,10, \text { and } 12)\) and only five possible odd values \((3,5,7,9, \text { and } 11),\) the dice should show even values more often than odd values. Do you agree with this statement? Explain. f. What method did you use to assign the probabilities requested?
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