How to find the possible the number of possible outcomes in the sample space and events?

Question

I need help with this problem:

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

• in the sample space?

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

• in the event that none of the team members considers himself or herself an Independent?

I know that I can star solving this by making a outcome tree or by the counting principle.

So by the counting principle, in the first one I can say that the first experiment is the type of work that has 2 outcomes and the second experiment is the political affiliation that has 3 possible outcomes, thus the number of possible outcomes is calculated by $2\cdot 3=6$, but that is the number of possibilities of just one of them, so I thought that the number of total outcomes should be $6\cdot 15= 90$, but the answer is $6^{15}$, why?

I have the same problem calculating the other two parts.

Answer 1

Note the phrasing: "determining the type of job (blue or white) and political affiliation (R, D, I) of 15 members of a team." So, the experimental unit (or outcome) is a team, not a player. For example, here are two outcomes: $$P_1(B,R)P_2(W,R)P_3(B,I)...P_{15}(W,D)\\ P_1(B,D)P_2(B,D)P_3(B,D)...P_{15}(B,D)$$ Note that the second outcome is homogeneous, since all players are identical. The sample space will consist of all possible teams, in particular, all sorts of teams with homogenous and heterogeneous players.

• Now each player can have $2$ types of job and $3$ types of political affiliation, hence $2\cdot 3=6$ different characteristics. Since the players' characteristics are independent of each other, we use the multiplication rule to find $6\cdot 6\cdots 6=6^{15}$.

• The number of outcomes in the event that at least one of the team members is a blue-collar worker can be found from the complement event, which includes the team members who are all white-collar. Since with job type is clear, there are only $3$ types of political affiliation for each member. So, it is $3^{15}$. By subtracting this number from the total number of outcomes in the sample space we find the answer: $6^{15}-3^{15}$.

• The number of outcomes in the event that none of the team members considers himself or herself an Independent is similar to previous. Can you find it?

Answer:

$(2\cdot 2)^{15}$

Answer 2

You need to use multiplication while you're on the same case, just like you did with the 2 options and 3 options: $2\cdot 3$. You need to use adding when there are different cases. you used adding 6 + 6 + ... + 6 = $6\cdot 15$ while you were on the same case.

For example, in the first case, you have 6 options for each member, so you need to multiply the options: $6\cdot 6\cdot ...\cdot 6 = 6^{15}$

Answer 3

You are determining the type of work and political affiliation of 15 people. The result of the of the experiment will be a $15$-tuple where the $n$th entry indicates the outcome for the $n$th person, $n=1,2,\dots,15.$ There are $6$ possible value for each entry, as you have shown.

How many possible $15$-tuples are there? There are $6$ values for the first entry. For each of these, there are $6$ values for the second enter, which gives us $36$ possibilities for the first $2$ entries. For each of these, there are $6$ possibilities for the third entry, and so on. In all, there are $6^{15}$ possible outcomes.

If $A,B$ are finite sets, then the cardinality of $A\times B$ is the cardinality of $A$ times the cardinality of $B$, right?