Question

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-causedinjuries 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 weremade 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? (4%)
b. What is the probability a person admitted to the hospital will die from a treatment-caused injury? (4%)
c. In the case of a negligent treatment-caused injury, what is the probability a malpractice claim will be paid? (4%)

Answer 1

General guidance

Concepts and reason

Probability:

The ratio of the number of favorable outcomes to certain event and total number of possible outcomes is called as the probability of an event.

Mutually Exclusive events:

If two events cannot occur or happen at the same time then events are said to be mutually exclusive. It is also known as disjoint event.

Non-mutually exclusive events:

If two events can occur or happen at the same time then events are said to be non-mutually exclusive.

Intersection of two events:

The set of the outcomes that belong to both the two events is called as intersection of two events.

Independent Events:

Two events are said to be independent events if the occurrence of one event does not affect the occurrence of another event. In other words, if occurring of an event A does not influence the probability of the event B occurring, then the two events A and B are independent.

Fundamentals

The probability of an event is defined as,

Probability Number of favorable outcomes for an event Total number of outcomes N(E Eis the event and Sis the sample spae] N(S)

Multiplication rule for independent events:

Two events A, B are said to be independent, if $P\\left( {A \\cap B} \\right) = P\\left( A \\right)P\\left( B \\right)$

First Step | All Steps | Answer Only

Step-by-step

Step 1 of 3

The probability a person admitted to the hospital will suffer a treatment-caused injury due to negligence is obtained as shown below:

Let A be the injuries, B be the death and D be the negligence.

From the given information, New York State found that 4% of the admission led to treatment \u2013caused injuries that is $P\\left( A \\right) = 0.04$. One seventh of these treatment-caused injuries resulted in death that is $P\\left( B \\right) = \\frac{1}{7}$, one-fourth were caused by negligence that is $P\\left( C \\right) = \\frac{1}{4}$

Let E be the Malpractice and F be the payments.

Malpractice claims were filed in one out of 7.5 cases involving negligence that is $P\\left( E \\right) = \\frac{1}{{7.5}}$, and payments were made in one out of every two claims that is $P\\left( F \\right) = \\frac{1}{2}$

The required probability is,

$\\begin{array}{c}\\\\P\\left( {A \\cap C} \\right) = P\\left( A \\right)P\\left( C \\right)\\\\\\\\ = 0.04 \\times \\frac{1}{4}\\\\\\\\ = 0.04 \\times 0.25\\\\\\\\ = 0.01\\\\\\end{array}$

The probability a person admitted to the hospital will suffer a treatment-caused injury due to negligence is 0.01.

Explanation | Hint for next step

The probability a person admitted to the hospital will suffer a treatment-caused injury due to negligence is obtained by multiplying the probability of injuries and negligence because the two are independently distributed. It is expected about 1% of times that the person admitted to the hospital will suffer a treatment-caused injury due to negligence.

Step 2 of 3

(b)

The probability a person admitted to the hospital will die from a treatment-caused injury.is obtained as shown below:

From the given information, New York State found that 4% of the admission led to treatment \u2013caused injuries that is $P\\left( A \\right) = 0.04$. One seventh of these treatment-caused injuries resulted in death that is $P\\left( B \\right) = \\frac{1}{7}$,

The required probability is,

$\\begin{array}{c}\\\\P\\left( {A \\cap B} \\right) = P\\left( A \\right)P\\left( B \\right)\\\\\\\\ = 0.04 \\times \\frac{1}{7}\\\\\\\\ = 0.04 \\times 0.1429\\\\\\\\ = 0.0057\\\\\\end{array}$

Part b

The probability a person admitted to the hospital will die from a treatment-caused injury.is 0.0057.

Explanation | Hint for next step

The probability a person admitted to the hospital will die from a treatment-caused injury.is obtained by multiplying the probability of injuries with the probability of death because the two are independently distributed. It is expected about 0.57% of times that a person admitted to the hospital will die from a treatment-caused injury

Step 3 of 3

(c)

The probability a malpractice claim will be paid is obtained as shown below:

Let E be the Malpractice and F be the payments.

Malpractice claims were filed in one out of 7.5 cases involving negligence that is $P\\left( E \\right) = \\frac{1}{{7.5}}$, and payments were made in one out of every two claims that is $P\\left( F \\right) = \\frac{1}{2}$

The required probability is,

$\\begin{array}{c}\\\\P\\left( {E \\cap F} \\right) = P\\left( E \\right)P\\left( F \\right)\\\\\\\\ = \\frac{1}{{7.5}} \\times \\frac{1}{2}\\\\\\\\ = 0.1333 \\times 0.50\\\\\\\\ = 0.0667\\\\\\end{array}$

Part c

The probability a malpractice claim will be paid is 0.0667.

Explanation

The probability a malpractice claim will be paid is obtained by multiplying the probability of malpractice and payments because the two are independently distributed. It is expected about 6.67% of times that malpractice claim will be paid.

Answer

The probability a person admitted to the hospital will suffer a treatment-caused injury due to negligence is 0.01.

Part b

The probability a person admitted to the hospital will die from a treatment-caused injury.is 0.0057.

Part c

The probability a malpractice claim will be paid is 0.0667.

Answer only

The probability a person admitted to the hospital will suffer a treatment-caused injury due to negligence is 0.01.

Part b

The probability a person admitted to the hospital will die from a treatment-caused injury.is 0.0057.

Part c

The probability a malpractice claim will be paid is 0.0667.

Answer 2

the answer for part c should be 0.00067