Mathematics · Ch 14 — Probability

Sample Space (Set Representation)

The sample space of a random experiment is the set of all possible outcomes of that experiment. It is usually denoted by $S$ , and each individual outcome is called an element, or a sample point, of $S$ .

S = \{\,\text{all possible outcomes of the random experiment}\,\}

Writing a Sample Space in Roster Form

For simple experiments, $S$ is written by listing every outcome inside braces (roster form), exactly as sets were represented in the earlier chapter on Sets.

One coin: $S = \{H, T\}$ , so $n(S) = 2$ .

One die: $S = \{1, 2, 3, 4, 5, 6\}$ , so $n(S) = 6$ .

Two coins tossed together: each coin independently shows H or T, so $S = \{HH, HT, TH, TT\}$ , $n(S) = 4$ . Note that $HT$ (first coin Head, second coin Tail) is a different outcome from $TH$ — the two coins are treated as distinguishable, exactly as the two dice below are.

Sample Space of Two Dice — a Systematic (Set-Builder) Construction

When two dice are thrown together, an outcome is an ordered pair $(x, y)$ , where $x$ is the number on the first die and $y$ is the number on the second die. In set-builder form,

S = \{(x, y) : x, y \in \{1, 2, 3, 4, 5, 6\}\}.

Since $x$ has $6$ possible values and, independently, $y$ has $6$ possible values, the multiplication principle gives $n(S) = 6 \times 6 = 36$ .

Watch out

$(x, y)$ and $(y, x)$ are different sample points whenever $x \ne y$ — do not collapse $(2,5)$ and $(5,2)$ into a single outcome. Treating the sample space as unordered pairs under-counts $n(S)$ and gives wrong probabilities later.

Sample Spaces Built From a Restriction …

Misc 1Sample space for two dice thrown together — systematic (row, column) structure

Worked out. The sample space for throwing two dice together consists of all ordered pairs (x, y) where x is the number shown on the first die (1 through 6) and y is the number shown on the second die (1 through 6). It is built systematically as a 6-row-by-6-column grid: Row 1 (first die shows 1): (1,1), (1,2), (1,3), (1,4), (1,5), (1,6). Row 2 (first die shows 2): (2,1), (2,2), (2,3), (2,4), (2,5), (2,6). Row 3 (first die shows 3): (3,1), (3,2), (3,3), (3,4), (3,5), (3,6). Row 4 (first die shows 4): (4,1), (4,2), (4,3), (4,4), (4,5), (4,6). Row 5 (first die shows 5): (5,1), (5,2), (5,3), (5,4), (5,5), (5,6). Row 6 (first die shows 6): (6,1), (6,2), (6,3), (6,4), (6,5), (6,6). Each row contains exactly 6 ordered pairs and there are 6 rows, giving a total of 6 x 6 = 36 equally likely outcomes. The pair (x,y) is treated as different from (y,x) whenever x is not equal to y, since the two dice are distinguishable outcomes of a single throw -- so (1,2) and (2,1) are two separate sample points, both present in the sample space. The 6 outcomes lying on the main diagonal, where x = y -- n …

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