Chapter 2.5, Problem 34E

Patients arriving at a hospital outpatient clinic can select one of three stations for service. Suppose that physicians are assigned randomly to the stations and that the patients therefore have no station preference. Three patients arrive at the clinic and their selection of stations is observed.

1. a List the sample points for the experiment.
2. b Let A be the event that each station receives a patient. List the sample points in A.
3. c Make a reasonable assignment of probabilities to the sample points and find P(A).

To determine

Provide the list of sample points for the given experiment.

Answer to Problem 34E

The list of sample points for the given experiment are as follows:

$S=\left\{\begin{array}{l}\left(s_1{s}_1{s}_1\right),\left(s_1{s}_1{s}_2\right),\left(s_1{s}_1{s}_3\right),\left(s_1{s}_2{s}_1\right),\left(s_1{s}_2{s}_2\right),\left(s_1{s}_2{s}_3\right),\left(s_1{s}_3{s}_1\right),\left(s_1{s}_3{s}_2\right),\left(s_1{s}_3{s}_3\right),\\ \left(s_2{s}_1{s}_1\right),\left(s_2{s}_1{s}_2\right),\left(s_2{s}_1{s}_3\right),\left(s_2{s}_2{s}_1\right),\left(s_2{s}_2{s}_3\right),\left(s_2{s}_3{s}_1\right),\left(s_2{s}_3{s}_2\right),\left(s_2{s}_3{s}_3\right),\left(s_2{s}_2{s}_2\right),\\ \left(s_3{s}_1{s}_1\right),\left(s_3{s}_1{s}_2\right),\left(s_3{s}_1{s}_3\right),\left(s_3{s}_2{s}_1\right),\left(s_3{s}_2{s}_2\right),\left(s_3{s}_2{s}_3\right),\left(s_3{s}_3{s}_1\right),\left(s_3{s}_3{s}_2\right),\left(s_3{s}_3{s}_3\right)\end{array}\right\}$

Explanation of Solution

Sample points:

When performing an experiment, it results with one or more outcomes. The possible outcomes in an experiment are called sample points. It is denoted by S.

Note that the patients can select one of the three stations. Denote $s_1,s_2\text{and}{s}_3$ as station 1, station 2, and station 3 respectively. Thus, each sample point of three patients has three-tuples.

The list of sample points for the given experiment is given below:

$S=\left\{\begin{array}{l}\left(s_1{s}_1{s}_1\right),\left(s_1{s}_1{s}_2\right),\left(s_1{s}_1{s}_3\right),\left(s_1{s}_2{s}_1\right),\left(s_1{s}_2{s}_2\right),\left(s_1{s}_2{s}_3\right),\left(s_1{s}_3{s}_1\right),\left(s_1{s}_3{s}_2\right),\left(s_1{s}_3{s}_3\right),\\ \left(s_2{s}_1{s}_1\right),\left(s_2{s}_1{s}_2\right),\left(s_2{s}_1{s}_3\right),\left(s_2{s}_2{s}_1\right),\left(s_2{s}_2{s}_3\right),\left(s_2{s}_3{s}_1\right),\left(s_2{s}_3{s}_2\right),\left(s_2{s}_3{s}_3\right),\left(s_2{s}_2{s}_2\right),\\ \left(s_3{s}_1{s}_1\right),\left(s_3{s}_1{s}_2\right),\left(s_3{s}_1{s}_3\right),\left(s_3{s}_2{s}_1\right),\left(s_3{s}_2{s}_2\right),\left(s_3{s}_2{s}_3\right),\left(s_3{s}_3{s}_1\right),\left(s_3{s}_3{s}_2\right),\left(s_3{s}_3{s}_3\right)\end{array}\right\}$

Thus, there are 27 sample points for the given experiment.

To determine

Provide the sample points of the event A.

Answer to Problem 34E

The sample points of the event A are as follows:

$A=\left\{\left(s_1{s}_2{s}_3\right),\left(s_1{s}_3{s}_2\right),\left(s_2{s}_1{s}_3\right),\left(s_2{s}_3{s}_1\right),\left(s_3{s}_1{s}_2\right),\left(s_3{s}_2{s}_1\right)\right\}$

Explanation of Solution

From Part (a), it is observed that there are 27 sample points in the sample space.

Among the 27 sample points, the sample points that each station receives a patient (event A) are as follows:

$A=\left\{\left(s_1{s}_2{s}_3\right),\left(s_1{s}_3{s}_2\right),\left(s_2{s}_1{s}_3\right),\left(s_2{s}_3{s}_1\right),\left(s_3{s}_1{s}_2\right),\left(s_3{s}_2{s}_1\right)\right\}$

Thus, there are 6 sample points in event A.

To determine

Give probability for each simple event.

Compute the probability for the event A, $P\left(A\right)$.

Answer to Problem 34E

The probability of each simple event is 0.0625.

The probability for the event A is 0.2222.

Explanation of Solution

Probability of an event:

To find the probability of an event, the following steps have to be carried out.

1. The experiment and the simple events should be clearly determined.

2. List out all the simple events associated with the experiment. This is known as sample space, S.

3. Assign probability for each of the simple events in S. Ensure that $P\left(E_i\right)\ge 0$ and ${\displaystyle \sum P\left(E_i\right)=1}$.

4. State the event of interest.

5. Obtain $P\left(A\right)$ by adding the probabilities of sample points in A.

From Part (a), there are 27 sample points in the sample space. Since the physicians are assigned randomly to the stations, each of the sample points are equally likely.

The probability of each simple event is computed as follows:

$\begin{array}{c}P\left(eachsimpleevent\right)=\frac{1}{27}\\ =0.037\end{array}$

That is, the probability of each simple event is 0.037.

From Part (b), it is clear that there are 6 sample points in event A.

The probability of the event A that each station receives a patient is computed as follows:

$\begin{array}{c}P\left(A\right)=\frac{6}{27}\\ =0.2222\end{array}$

Therefore, the probability of the event A is 0.2222.