Question

questions exercise 04.49 algo (conditional probability) a study of 31,000 hospital admissions in new york state found that 5% 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 (to 2 decimals)? b. what is the probability a person admitted to the hospital will die from a treatment - caused injury (to 3 decimals)? c. what is the probability a person admitted to the hospital will result in a malpractice claim that must be paid due to a negligent treatment that caused injury (to 5 decimals)?

Explanation:

Step1: Calculate probability of treatment - caused injury due to negligence

Let the total number of admissions be considered as 1. The proportion of treatment - caused injuries is 0.05. One - fourth of these treatment - caused injuries were caused by negligence. So the probability $P(N)$ (where $N$ is treatment - caused injury due to negligence) is $P(N)=0.05\times\frac{1}{4}$.
$P(N)=0.05\times\frac{1}{4}= 0.0125\approx0.01$

Step2: Calculate probability of death from treatment - caused injury

The proportion of treatment - caused injuries is 0.05. One - seventh of these treatment - caused injuries resulted in death. So the probability $P(D)$ (where $D$ is death from treatment - caused injury) is $P(D)=0.05\times\frac{1}{7}$.
$P(D)=0.05\times\frac{1}{7}\approx0.007$

Step3: Calculate probability of malpractice claim that must be paid

We know from Step1 that the probability of treatment - caused injury due to negligence is $P(N) = 0.0125$. Malpractice claims were filed in one out of 7.5 cases involving negligence, and payments were made in one out of every two claims. So the probability $P(C)$ (where $C$ is malpractice claim that must be paid) is $P(C)=0.0125\times\frac{1}{7.5}\times\frac{1}{2}$.
$P(C)=0.0125\times\frac{1}{7.5}\times\frac{1}{2}=\frac{0.0125}{15}\approx0.00083$

Answer:

a. 0.01
b. 0.007
c. 0.00083