Q. 2.7 - A First Course in Probability

Question

Consider an experiment that consists of determining the type of job—either blue collar or white collar— and the political affiliation—Republican, Democratic, or Independent—of the 15 members of an adult soccer team.

How many outcomes are

(a) in the sample space?

(b) in the event that at least one of the team members is a blue-collar worker?

(c) in the event that none of the team members considers himself or herself an Independent?

Step-by-Step Solution

Answer

(a) The outcomes in the sample space are $6^{15}$.

(b) The outcomes in the event that at least one of the team members is a

blue-collar worker are $6^{15}-3^{15}$.

(c) The outcomes in the event that none of the team members considers

himself or herself an Independent are 

Part (a) Step 1. Find the outcomes in the sample space.

Each member can have either two types of jobs - blue collared or white collared.

Further each member can be either Republican or Democratic or Independent.

So, there are three options. That means every member can have one of the $2\times 3=6$ categories.

The no. of members are $15$.

Hence, the sample space is $6^{15}$.

Part (b) Step 1. Find the outcomes in the event that at least one of the team members is a blue-collar worker.

The possible no. of cases for the event that at least one of the team members is a blue collar worker means, we will exclude the cases that all the members are not blue collar from the total sample space.

So, we will consider the level of job type with only blue collar.

This means $6^{15}-{\left(1\times 3\right)}^{15}$.

Therefore, the outcomes in the event that at least one of the team members is a blue-collar worker are $6^{15}-3^{15}$.

Part (c) Step 1. Find the outcomes in the event that none of the team members considers himself or herself an Independent. .

The event that none of the team members considers himself or herself an independent means the $15$ members are from either Republican or Democratic Party.

Therefore, the number of outcomes per members is $2\times 2=4$

The total members are $15$

Therefore, the total number of possible outcomes are $4^{15}$.