I’m revising my stats fundamentals (trying to shore up the basics!), and I keep getting tangled on sample spaces. Example: roll two fair six-sided dice and record the sum. My brain keeps doing a little loop-de-loop here.
In one set of notes, the sample space is written as {2,3,4,5,6,7,8,9,10,11,12}. In another, it’s all 36 ordered pairs like (1,1), (1,2), …, (6,6). I’m not sure which one I’m actually supposed to write for “the” sample space of this experiment.
Here’s my completely wrong attempt: I wrote S = {2,3,4,5,6,7,8,9,10,11,12} and then I gave each outcome probability 1/11 because there are 11 sums. That felt so tidy at 11pm and now it just feels… wrong.
So: what is the correct sample space for “roll two dice and look at the sum,” and how do I decide in general? Is it context-dependent (like, are we modeling the physical outcomes versus just the sum), or is one of these just not a valid sample space for this situation?
Bonus mini-confusion: for drawing two cards without replacement from a standard deck, should the sample space be ordered pairs of specific cards, or can it be something like all 2-card combinations, or even just counts like “number of hearts drawn”? I keep mixing up sample spaces with the thing I’m measuring.
I’m trying to strengthen my fundamentals, so a simple way to decide “what goes in S” would help me a lot. Any help appreciated!
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3 Responses
Pick S to match what you “see”: if you only record the sum, S = {2,…,12} (not equally likely), but if you watch the dice themselves, S = {(i,j): i,j∈{1,…,6}} with 36 equally likely outcomes; cards are the same idea (ordered pairs, 2-card combinations, or just counts like {0,1,2} for hearts). Example: P(sum = 7) = 6/36 because the pairs (1,6),(2,5),(3,4),(4,3),(5,2) are the ones that do it-six of them, I promise, even if my eyeballs only spot five before tea.
I think of sample spaces like choosing the “resolution” of a photo. If I roll two dice, the full-resolution picture is the 36 ordered pairs (1,1) through (6,6) – every pixel equally likely, super crisp, easy to count. Then the sum is like applying a blur filter: many crisp pixels funnel into the same brightness level (the sums 2–12). Both are valid sample spaces; they just answer slightly different questions. If the experiment is literally “record the sum,” you can absolutely take S = {2,3,4,5,6,7,8,9,10,11,12}. But the probabilities aren’t uniform there: P(7) = 6/36, P(6) = 5/36, P(2) = 1/36, and so on, because those come from how many ordered pairs land on each sum. So your 1/11 instinct felt tidy (I’ve done that late at night too!), but the sums aren’t equally likely. I’m pretty sure the safest workflow is: start with the 36 equally likely pairs, then “push forward” to sums by counting how many pairs give each sum.
General rule of thumb I use (and maybe I oversell it a tad): pick a sample space at the level of detail you’re actually distinguishing, ideally one where the outcomes are equally likely – that’s basically the definition of a good sample space in practice. Then anything you “measure” (like the sum) is a function on that space. If later you decide you only care about the sum, you can always coarsen the picture without changing the final probabilities for that summary.
For the card bonus: drawing two without replacement can be modeled as ordered pairs of distinct cards (52×51 outcomes, all equally likely), or as 2-card hands ignoring order (C(52,2) outcomes, also equally likely). Both are fine – choose the one that matches what you’ll ask next. If you only care about “number of hearts,” you can even use S = {0,1,2} with P(0) = C(39,2)/C(52,2), P(1) = C(13,1)C(39,1)/C(52,2), P(2) = C(13,2)/C(52,2). I’m Ninety-something percent confident that thinking in terms of “start detailed and then compress to what you report” will keep you out of the loop-de-loops.
Both sample spaces you saw are valid; they’re just modeling different levels of detail. If you take the 36 ordered pairs, each has probability 1/36, and “sum” is a random variable on that space. If you only care about the sum, you can compress the space to {2,3,…,12}, but the probabilities are then induced, not equal: P(2)=1/36, P(3)=2/36, …, P(7)=6/36, then symmetrically back down to P(12)=1/36. I once wrote 1/11 for each sum too-very tidy, and very wrong-until I counted how many pairs make 7 versus 2. That picture of a little triangle of counts fixed it for me.
A good rule: choose a sample space fine enough to distinguish everything you might ask about, and then define the quantity you “record” as a function (a random variable) on that space. If you later decide you only care about that function, you can push the probabilities forward to a smaller space. Neither choice is “the” space; consistency between S, its probabilities, and the events you analyze is what matters.
For two cards without replacement: if order matters (first vs second), use ordered pairs of distinct cards; all such pairs have probability 1/(52·51). If only the 2-card hand matters, use the 52-choose-2 subsets; each hand then has probability 1/C(52,2). Counts like “number of hearts” are random variables on either space, and their probabilities follow from whichever space you picked. I keep it simple: pick the smallest space that still lets me answer the questions I care about without extra bookkeeping.