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.
Answers
Step-by-step explanation:
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,
ProbabilityNumber of favorable outcomes for an eventTotal number of outcomesN(EEis 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)P(A\u2229B)=P(A)P(B)
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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) = 0.04P(A)=0.04. One seventh of these treatment-caused injuries resulted in death that is P\\left( B \\right) = \\frac{1}{7}P(B)=
7
1
\u200b , one-fourth were caused by negligence that is P\\left( C \\right) = \\frac{1}{4}P(C)=
4
1
\u200b
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}}P(E)=
7.5
1
\u200b , and payments were made in one out of every two claims that is P\\left( F \\right) = \\frac{1}{2}P(F)=
2
1
\u200b
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}
P(A\u2229C)=P(A)P(C)
=0.04\u00d7
4
1
\u200b
=0.04\u00d70.25
=0.01
\u200b
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) = 0.04P(A)=0.04. One seventh of these treatment-caused injuries resulted in death that is P\\left( B \\right) = \\frac{1}{7}P(B)=
7
1
\u200b ,
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}
P(A\u2229B)=P(A)P(B)
=0.04\u00d7
7
1
\u200b
=0.04\u00d70.1429
=0.0057
\u200b
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}}P(E)=
7.5
1
\u200b , and payments were made in one out of every two claims that is P\\left( F \\right) = \\frac{1}{2}P(F)=
2
1
\u200b