Q. 2.6 - 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) 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

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 $S$  $=\left\{\left(1,g\right), \left(1,f\right), \left(1,s\right), \left(0,g\right), \left(0,f\right), \left(0,s\right)\right\}$

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 be  $\left\{\left(1,s\right), \left(0,s\right)\right\}$

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 be  $\left\{\left(0.g\right), \left(0,f\right), \left(0,s\right)\right\}$

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

We have,

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