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 $B^c\cup A$.

Short answer

(a) The sample space of this experiment are$=\left\{\left(1,g\right),\left(1,f\right),\left(1,s\right),\left(0,g\right),\left(0,f\right),\left(0,s\right)\right\}$

(b) The outcomes in $A$are$\left\{\left(1,s\right),\left(0,s\right)\right\}$

(c) The outcomes in $B$are$\left\{\left(0.g\right),\left(0,f\right),\left(0,s\right)\right\}$

(d) The outcomes in the event $B^c\cup A$are

Step-by-step solution

Part (a) Step 1. Construct the sample space 

The information given is related to data on incoming patients into the hospital.

According to the given information,

Let $1$be that the patient has insurance and $0$be that the patient do no has insurance.

Let the condition of patient be -

Good - $g$

Fair - $f$

Serious - $s$

Therefore, the sample space $S$localid="1649317437417" role="math" $=\left\{\left(1,g\right),\left(1,f\right),\left(1,s\right),\left(0,g\right),\left(0,f\right),\left(0,s\right)\right\}$

Part (b) Step 1. Compute the sample space of A.

Let $1$ be that the patient has insurance and $0$ be that the patient do no has insurance.

Let the condition of patient be -

Good - $g$

Fair - $f$

Serious - $s$

The event $A$represents the patients in serious condition.

So the outcomes in$A$will berole="math" localid="1649317522513" $\left\{\left(1,s\right),\left(0,s\right)\right\}$

Part (c) Step 1. Compute the sample space of B.

Let $1$be that the patient has insurance and $0$be that the patient do no has insurance.

Let the condition of patient be -

Good - $g$

Fair - $f$

Serious - $s$

The event $B$represents the uninsured patients.

So the outcomes in $B$will berole="math" localid="1649317460339" $\left\{\left(0.g\right),\left(0,f\right),\left(0,s\right)\right\}$

Part (d) Step 1. Compute the sample space of Bc∪A

We have,

$$\begin{gathered} S=\left\{\left(1,g\right),\left(1,f\right),\left(1,s\right),\left(0,g\right),\left(0,f\right),\left(0,s\right)\right\} \\ B=\left\{\left(0.g\right),\left(0,f\right),\left(0,s\right)\right\} \\ A=\left\{\left(1,s\right),\left(0,s\right)\right\} \\ \Rightarrow B^c=\left\{\left(1,g\right),\left(1,f\right),\left(1,s\right)\right\} \\ {B}^c\cup A=\left\{\left(1,g\right),\left(1,f\right),\left(1,s\right),\left(0,s\right)\right\} \end{gathered}$$