91Ó°ÊÓ

Medical insurance status-covered (C) or not covered (N) - is determined for each individual arriving for treatment at a hospital's emergency room. Consider the chance experiment in which this determination is made for two randomly selected patients. The simple events are \(O_{1}=(\mathrm{C}, \mathrm{C})\), meaning that the first patient selected was covered and the second patient selected was also covered, \(O_{2}=(\mathrm{C}, \mathrm{N}), O_{3}=(\mathrm{N}, \mathrm{C})\), and \(O_{4}=\) \((\mathrm{N}, \mathrm{N}) .\) Suppose that probabilities are \(P\left(O_{1}\right)=.81\), \(P\left(O_{2}\right)=.09, P\left(O_{3}\right)=.09\), and \(P\left(O_{4}\right)=.01\). a. What outcomes are contained in \(A\), the event that at most one patient is covered, and what is \(P(A)\) ? b. What outcomes are contained in \(B\), the event that the two patients have the same status with respect to coverage, and what is \(P(B)\) ?

Step by step solution

01

Identify Outcomes of Event A

Event A is defined as 'at most one patient is covered'. The outcomes that satisfy this condition are \((C, N)\), \((N, C)\), and \((N, N)\), corresponding to the situations where only the first patient is covered, only the second patient is covered, and neither patient is covered, respectively.

02

Calculate Probability of Event A

The probability of event A, \(P(A)\), is the sum of probabilities of outcomes \((C, N)\), \((N, C)\), and \((N, N)\), which are .09, .09, and .01 respectively. So, \(P(A) = P(O_2) + P(O_3) + P(O_4) = .09 + .09 + .01 = .19\)

03

Identify Outcomes of Event B

Event B is defined as 'the two patients have the same status with respect to coverage'. The outcomes that satisfy this condition are \((C, C)\) and \((N, N)\), corresponding to the situations where both patients are covered and neither patient is covered, respectively.

04

Calculate Probability of Event B

The probability of event B, \(P(B)\), is the sum of probabilities of outcomes \((C, C)\) and \((N, N)\), which are .81 and .01 respectively. So, \(P(B) = P(O_1) + P(O_4) = .81 + .01 = .82\)

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Key Concepts

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

Random Experiments

A random experiment is a process or action that produces uncertain results, and understanding them is essential for mastering probability theory. In the given exercise, the random experiment is performed to determine the insurance status of two patients admitted consecutively to the emergency room. Each patient can either be covered (C) or not covered (N). This is a basic example of a random experiment, where the outcome cannot be predicted with certainty before the experiment is conducted.

The key characteristics of a random experiment include:

• The possibility of multiple outcomes. In our case, there are four possible outcomes regarding the pair of patients: \((C, C), (C, N), (N, C), (N, N)\).
• Each outcome is determined by chance, meaning none is favored over the others just by looking at the experiment's setup.
• It can be repeated under similar conditions, though each repetition may yield a different result.

Understanding random experiments is crucial as they lay the foundation for further study into probability theory, where we analyze these different outcomes quantitatively.

Event Probability

Event probability refers to the likelihood of an event occurring within the scope of a random experiment. To understand this, let's define what an event is: an event is a set of outcomes. When you perform a random experiment, you can have several different events, each consisting of one or more outcomes.

In the exercise, we have two events of interest:

• Event A: At most one patient is covered. This event includes the outcomes \((C, N), (N, C),\) and \((N, N)\).
• Event B: Both patients have the same insurance status. This event includes the outcomes \((C, C)\) and \((N, N)\).

The probability of an event is calculated by summing the probabilities of all the specific outcomes that make up the event.

For example, for Event A, the probability is:
\[P(A) = P(O_2) + P(O_3) + P(O_4) = 0.09 + 0.09 + 0.01 = 0.19\]
Similarly, for Event B, we have:
\[P(B) = P(O_1) + P(O_4) = 0.81 + 0.01 = 0.82\]By computing these probabilities, we gain insight into how likely certain scenarios are in the context of our random experiment.

Outcomes in Probability

In probability theory, outcomes are the basic results of a random experiment. For any random experiment, there is a set of all possible outcomes known as the sample space. Each outcome is considered a 'simple event'. In this context, a simple event is one of the specific possible results of an experiment.

In the emergency room scenario, the possible outcomes are:

• \((C, C)\): Both patients are covered.
• \((C, N)\): The first patient is covered, but the second is not.
• \((N, C)\): The first patient is not covered, but the second is covered.
• \((N, N)\): Neither patient is covered.

Each of these outcomes has an associated probability, reflecting how likely each outcome is when performing the experiment.

Understanding outcomes is vital for determining the probabilities of events, since events are combinations of these outcomes. By establishing the probabilities for each outcome, we can compute the probability for other events of interest, analyzing which situations are more likely and allowing for informed decision-making based on probability theory.