Question

A study of hospital admissions in New York State found that 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 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 (to 2 decimals)? .01 b. What is the probability a person admitted to the hospital will die from a treatment caused injury (to 3 decimals)? .006 c. In the case of a negligent treatment-caused injury, what is the probability a malpractice claim will be paid(to 5 decimals)?

Answer 1

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

(b) The probability a person admitted to the hospital will die from a treatment caused injury is 0.006.

(c) In the case of a negligent treatment-caused injury the probability a malpractice claim will be paid is 0.06667.

Explanation:

Denote the events as follows:

X = treatment-caused injuries

A = injuries resulted in death

B = injuries were caused by negligence

C = Malpractice claims were filed

D = payments were made in cases where claims were filed.

The information provided is:

`P(X)=0.04\\P(A|X)=(1)/(7)\\P(B|X)=(1)/(4)\\P(C|B)=(1)/(7.5)\\P(D|C)=(1)/(2)`

The multiplication rule of probability is:

`P(U\cap V)=P(U|V)* P(V)=P(V|U)* P(U)`

(a)

Compute the value of P (X ∩ B) as follows:

`P(X\cap B)=P(B|X)P(X)=(1)/(4)* 0.04=0.01`

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

(b)

Compute the value of P (X ∩ A) as follows:

`P(X\cap A)=P(A|X)P(X)=(1)/(7)* 0.04=0.0057\approx0.006`

Thus, the probability a person admitted to the hospital will die from a treatment caused injury is 0.006.

(c)

Compute the value of P (C ∩ D | B) as follows:

`P(C\cap D|B)=P(C|B)* P(D|C)=(1)/(7.5)* (1)/(2)=0.06667`

Thus, in the case of a negligent treatment-caused injury the probability a malpractice claim will be paid is 0.06667.