Question

A family consisting of three persons—A, B, and C—goes to a medical clinic that always has a doctor at each of stations 1, 2, and 3. During a certain week, each memberof the family visits the clinic once and is assigned at random to a station. The experiment consists of recording the station number for each member. One outcome is (1, 2, 1) for Ato station 1, Bto station 2, and Cto station 1.

a. List the 27 outcomes in the sample space.

b. List all outcomes in the event that all three members go to the same station.

c. List all outcomes in the event that all members go to different stations.

d. List all outcomes in the event that no one goes to station 2.

Short answer

a. The sample space is given as,

\(S = \left\{ \begin{aligned}\left( {1,1,1} \right),\left( {1,1,2} \right),\left( {1,1,3} \right),\left( {1,2,1} \right),\left( {1,2,2} \right),\left( {1,2,3} \right),\left( {1,3,1} \right),\left( {1,3,2} \right),\left( {1,3,3} \right),\left( {2,1,1} \right),\left( {2,1,2} \right),\\\left( {2,1,3} \right),\left( {2,2,1} \right),\left( {2,2,2} \right),\left( {2,2,3} \right),\left( {2,3,1} \right),\left( {2,3,2} \right),\left( {2,3,3} \right),\left( {3,1,1} \right),\left( {3,1,2} \right),\left( {3,1,3} \right),\left( {3,2,1} \right)\\\left( {3,2,2} \right),\left( {3,3,1} \right),\left( {3,3,2} \right),\left( {3,3,3} \right)\end{aligned} \right\}\)

b. The possible outcomes are:

\(A = \left\{ {\left( {111} \right),\left( {222} \right),\left( {333} \right)} \right\}\)

c. The possible outcomes are:

\(B = \left\{ {\left( {1,2,3} \right),\left( {1,3,2} \right),\left( {2,1,3} \right),\left( {2,3,1} \right),\left( {3,1,2} \right),\left( {3,2,1} \right)} \right\}\)

d. The possible outcomes are:

\(D = \left\{ {\left( {1,1,1} \right),\left( {1,1,3} \right),\left( {1,3,1} \right),\left( {1,3,3} \right),\left( {3,1,1} \right),\left( {3,1,3} \right),\left( {3,3,1} \right),\left( {3,3,3} \right)} \right\}\)

Step-by-step solution

Given information

The number of persons in a family is 3; they are, A, B, and C.

They went to a medical clinic that has a doctor at each of stations 1, 2, and 3.

Each member visits the clinic once and is assigned the station number.

The outcome of an experiment (1,2,1) for A to station 1, B to station 2, and C to station 1.

List of the possible outcomes in sample space

a.

Let S represents the sample space.

The sample space for the given experiment is,

\(S = \left\{ \begin{aligned}\left( {1,1,1} \right),\left( {1,1,2} \right),\left( {1,1,3} \right),\left( {1,2,1} \right),\left( {1,2,2} \right),\left( {1,2,3} \right),\left( {1,3,1} \right),\left( {1,3,2} \right),\left( {1,3,3} \right),\left( {2,1,1} \right),\left( {2,1,2} \right),\\\left( {2,1,3} \right),\left( {2,2,1} \right),\left( {2,2,2} \right),\left( {2,2,3} \right),\left( {2,3,1} \right),\left( {2,3,2} \right),\left( {2,3,3} \right),\left( {3,1,1} \right),\left( {3,1,2} \right),\left( {3,1,3} \right),\left( {3,2,1} \right)\\\left( {3,2,2} \right),\left( {3,3,1} \right),\left( {3,3,2} \right),\left( {3,3,3} \right)\end{aligned} \right\}\)

List of the possible outcomes

b.

Let A represents the event that all three members go to the same station.

The possible numbers of outcomes are,

\(A = \left\{ {\left( {111} \right),\left( {222} \right),\left( {333} \right)} \right\}\)

List of the possible outcomes

c.

Let B represents the event that all members go to different stations.

The possible outcomes of event B is represented as,

\(B = \left\{ {\left( {1,2,3} \right),\left( {1,3,2} \right),\left( {2,1,3} \right),\left( {2,3,1} \right),\left( {3,1,2} \right),\left( {3,2,1} \right)} \right\}\)

List of the possible outcomes

d.

Let D represents the event that no one goes to station 2.

The possible outcomes for the event D is,

\(D = \left\{ {\left( {1,1,1} \right),\left( {1,1,3} \right),\left( {1,3,1} \right),\left( {1,3,3} \right),\left( {3,1,1} \right),\left( {3,1,3} \right),\left( {3,3,1} \right),\left( {3,3,3} \right)} \right\}\)