consider an experiment that consists of determining the type of job – either blue-collar or white collar- and the political affiliation -republicans, democratic or independent – of the 15 members of an adult soccer team. how many outcomes are in the event that at least one of the team members is a blue-collow worker?
Why is the answer 6^15 – 3^15?
why can't it be 3 * 6^14
Best Answer
Our experiment proceeds as follows. We line up the team members, say by Social Security number. We now ask each in turn her/his job description and political affiliation, and record the information.
For any person, the information can be recorded as an ordered pair $(x,y)$, where $x$ is one of $B$ or $W$, and $y$ is one of $R$, $D$, or $I$. So there are for any person $6$ possible records. The team outcome is a sequence of such records, of length $15$. Thus for our team there are $6^{15}$ possible outcomes.
How many of these outcomes have at least one blue-collar worker? How many have no blue-collar worker? Then $x=W$, and $y$ can still take on any of $3$ values, for a total of $3^{15}$ possibilities. Thus the number of possible summaries with at least one $B$ is $6^{15}-3^{15}$.
Remark: What does your answer of $(3)(6^{14})$ count? It seems to count the number of ways that the first person asked is a $B$, and all the rest are free to be anything. Certainly in that case there will be at least one $B$. But there can be at least one $B$ in other ways. We could try to do a fancier version of your idea, but then we can end up double-counting, or worse. That can be compensated for by using the Method of Inclusion/Exclusion, but it gets complicated.