If we define the population as the complete set of items or "events" of interest.
And
We define the sample space as the set of all possible outcomes (exhaustively) from a random experiment.
Then, I wondered this...
Take a dice roll. The population is the complete set of possible items {1, 2, 3, 4, 5, 6}. The sample space is the set of all possible outcomes, also {1, 2, 3, 4, 5, 6}. So here sample space and population appear to be the same thing, so when are they not and what are the distinguishing factors between the two??
The WikiPedia page on sample spaces caused the penny to drop for me:
...For many experiments, there may be more than one plausible sample
space available, depending on what result is of interest to the
experimenter. For example, when drawing a card from a standard deck of
fifty-two playing cards, one possibility for the sample space could be
the various ranks (Ace through King), while another could be the suits
(clubs, diamonds, hearts, or spades)...
Ah ha! So my population is the set of all cards {1_heart, 2_heart, ..., ace_heart, 1_club, ...} but the sample space may be, if we are looking for the suits, just {heart, club, diamond, spade}. So the population and sample space are different here.
In summary the population is the set of items I'm looking at. The sample space may or may not be the population... that depends on what question about the population is being asked.
This answers the latter half of my question... possibly I didn't ask it clearly enough or it was just too obvious (it just took a while for it to sink into my head)
The other half of the question is answered by "Qwerty". In all the sources I've looked at it appears classical probability treats events as equally likely. [1] [2] (and the book I referenced in the Q). "Qwerty" has expanded it slightly... but I believe this is where relative frequency probability comes into play and allows us to model "unfair" (not equally likely) events: From [2]
The probability of an event is the ratio of the number of cases
favorable to it, to the number of all cases possible when nothing
leads us to expect that any one of these cases
should occur more than any other, which renders them, for us, equally
possible.
And
The classical definition of probability was called into question by
several writers of the nineteenth century, including John Venn and
George Boole.2 The frequentist definition of probability became
widely accepted as a result of their criticism
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?
Best Answer
You are correct about the “experiment” — it is drawing 4 apples (without replacement I assume).
Given this experiment, the sample space is the set of possible outcomes.
E.g., $GGGG, RGGG, RRGG, ...$