Problem 6 - A First Course in Probability

Question

A hospital administrator codes incoming patients suffering gunshot wounds according to whether they have insurance (coding 1 if they do and 0 if they do not) and according to their condition, which is rated as good (g), fair (f), or serious (s). Consider an experiment that consists of the coding of such a patient. (a) Give the sample space of this experiment. (b) Let \(A\) be the event that the patient is in serious condition. Specify the outcomes in \(A\) (c) Let \(B\) be the event that the patient is uninsured. Specify the outcomes in \(B\). (d) Give all the outcomes in the event \(B^{c} \cup A\).

Step-by-Step Solution

Answer

(a) Sample Space: \(S = \{1g, 1f, 1s, 0g, 0f, 0s\}\) (b) Outcomes in A (Serious condition): \(A = \{1s, 0s\}\) (c) Outcomes in B (Uninsured): \(B = \{0g, 0f, 0s\}\) (d) Outcomes in \(B^c \cup A\): \(B^c \cup A = \{1g, 1f, 1s, 0s\}\)

Step 1: Coding the sample space

Each patient can be coded by combining two factors - insurance status (1 or 0) and condition (g, f, or s). For example, a patient with insurance and in good condition can be coded as 1g. Thus, possible patient codes can be: 1. 1g - Patient with insurance, good condition 2. 1f - Patient with insurance, fair condition 3. 1s - Patient with insurance, serious condition 4. 0g - Patient with no insurance, good condition 5. 0f - Patient with no insurance, fair condition 6. 0s - Patient with no insurance, serious condition These compose the sample space (S) of the experiment which can be written as: \(S = \{1g, 1f, 1s, 0g, 0f, 0s\}\)

Step 2: (a) Sample Space

The sample space of the experiment consists of all possible insurance status and condition of gunshot wound patients. Sample Space: \(S = \{1g, 1f, 1s, 0g, 0f, 0s\}\)

Step 3: (b) Outcomes in A (Serious condition)

Event A is defined as the patient being in serious condition (s), regardless of insurance status: \(A = \{1s, 0s\}\)

Step 4: (c) Outcomes in B (Uninsured)

Event B is defined as the patient being uninsured (0), regardless of their condition: \(B = \{0g, 0f, 0s\}\)

Step 5: (d) Outcomes in \(B^c \cup A\)

First, find the complement of event B (\(B^c\)), which is the event that the patient is insured: \(B^c = \{1g, 1f, 1s\}\) Now, we need the union of events \(B^c\) and A, which represents the outcomes in which either patient is insured or is in serious condition: \(B^c \cup A = \{1g, 1f, 1s, 0s\}\)

Key Concepts

Patient Coding in Healthcare

Understanding patient coding in healthcare is crucial for managing not only administrative details but also for statistical analysis and insurance processing. Patient coding is a way to categorize various aspects of a patient's experience in a healthcare facility. For gunshot wound patients, patient coding might include both the individual's insurance status and their health condition.

Codes such as '1' or '0' can represent whether a patient has insurance or not, with '1' indicating insured and '0' signifying uninsured. Health conditions could be encoded as 'g' for good, 'f' for fair, and 's' for serious. These alphanumeric codes streamline the process of documenting and tracking patient data. Hospitals use patient coding to facilitate billing, update medical records, and guide healthcare decisions.

Probability Events and Outcomes

In the context of probability, an event is a set of outcomes of an experiment (a process that leads to a result) to which a probability is assigned. For the coding of patients, events can be represented by the conditions such as having insurance or being in a specific health state.

The outcomes are the possible results of the experiment; when considering the coding of patients with gunshot wounds, outcomes include all combinations of insurance status and condition. These outcomes make up the sample space, a term that encompasses all possible results of a probability experiment. The sample space for the patient coding would be \(S = \{1g, 1f, 1s, 0g, 0f, 0s\}\).

Complementary Events in Probability

Within probability theory, the complementary event mathematically represents 'everything in the sample space that is not within the specified event'. For a set event \(B\), its complement, denoted as \(B^c\), would include all outcomes that do not fit the criteria for \(B\).

In the exercise related to patient conditions, if event \(B\) represents uninsured patients, then \(B^c\) represents all the outcomes where patients have insurance, that is, \(B^c = \{1g, 1f, 1s\}\). Complementary events are a fundamental part of calculating probabilities as they often make it easier to calculate the likelihood of an event not happening rather than it happening.

Union of Events in Probability

The union of events in probability is a concept used to determine the likelihood of any one of several events occurring. Represented by the symbol \(\cup\), it combines the outcomes of multiple events into one set.

For example, when we are interested in the outcomes where a patient is insured or in serious condition (events \(B^c\) and \(A\)), the union of these events, \(B^c \cup A\), would include every outcome that is covered by either event. In this exercise, \(B^c \cup A = \{1g, 1f, 1s, 0s\}\). This union does not double-count the outcomes that appear in both events; it's like combining two lists and removing any duplicates to have a clear view of all unique possibilities.