Sample Space vs Distribution

The sample space, usually written $S$, is the complete list of outcomes of one trial. The list has to satisfy two demands at once: the outcomes must be mutually exclusive, so no two can happen together, and exhaustive, so at least one must happen. Meet both and exactly one outcome occurs on every trial, which is why their probabilities add to 1. An event is then any subset of $S$.

A bag holds one red chip, one blue and one green. Draw a chip, record the color, put it back, and draw again, keeping the order. Stages multiply, so $S$ holds $3 \times 3 = 9$ ordered pairs and each has probability $1/9 \approx 0.1111$. The event "the two draws match" holds 3 of the 9, giving $3/9 \approx 0.3333$.

"A sample space has to be a list of equally likely outcomes." It does not. Spin a wheel divided into a half labeled A, a quarter labeled B and a quarter labeled C. Then $S = \{A, B, C\}$ with probabilities 0.5, 0.25 and 0.25: exclusive, exhaustive, adding to 1, and a perfectly good sample space. Equal likelihood is a separate assumption, and what it buys you is the right to count instead of add. The requirement that actually binds is the total, so if your probabilities do not reach 1 you have left an outcome out, and that check is the cheapest way to catch the omission before it spoils everything downstream.

A sample space need not be finite, or even listable. Waiting for a bus that arrives at some point in the next ten minutes gives $S$ as every real number from 0 to 10. Nothing can be counted there, probability comes from area under a density curve, and each exact arrival time has probability 0 while the bus still arrives.

Writing $S$ down first is the habit that makes probability problems reliable, since every rule in Unit 2 is stated over one. Once a process runs in stages, a tree diagram is the usual way to keep the list complete without dropping a branch.

A distribution belongs to one variable and answers two questions at once: which values occur, and how often each one does. For a categorical variable that is the count or the share in every category. For a quantitative variable it is the pattern of values you read off a dotplot, histogram, or stemplot. For a random variable it is every possible value paired with its probability, and those probabilities have to sum to 1.

Twelve students report their number of siblings: 0, 0, 1, 1, 1, 1, 2, 2, 2, 3, 3, 5. The distribution is the whole tally: 0 occurs twice, 1 occurs four times, 2 occurs three times, 3 occurs twice, 4 occurs zero times, and 5 occurs once. That zero at 4 is part of it. The mean, $21/12 = 1.75$, and the median, 1.5, are summaries pulled out of the tally, not the distribution itself.

"Describe the distribution" answered with one number is the most expensive habit on the free-response section. "The mean is 1.75" describes nothing. A description names shape, center, spread, and any unusual features such as outliers, gaps, or clusters, in the context of the variable: skewed right, centered near 1.5 siblings, running from 0 to 5, with the single student who has 5 siblings standing apart from the rest.

One word covers three objects that beginners collapse into one: the distribution of a population, the distribution of a single sample drawn from it, and the sampling distribution of a statistic computed from many such samples. All three are distributions, they have different spreads, and a large share of the confusion in inference comes from reading a sentence about one as a sentence about another.

A distribution is also a one-variable idea. A scatterplot shows two variables at once, so on its own it displays no distribution; you would describe the distribution of each variable separately, or describe the relationship between them. Graphing and describing one quantitative variable are Unit 1 topics 1.5 and 1.6.