Sample Spaces: Discrete and Continuous

Before you can find a probability, list the possibilities

To find how likely something is, you first need to know everything that could happen. That complete list is the sample space, written S. Each item in it is an outcome.

We write sample spaces as sets, using braces:

• Toss a coin: S = {H, T}, so n(S) = 2.
• Roll a die: S = {1, 2, 3, 4, 5, 6}, so n(S) = 6.
• Draw one card from a standard deck: S has n(S) = 52 outcomes.

A standard deck has 52 cards in 4 suits: hearts ♥ and diamonds ♦ (red), clubs ♣ and spades ♠ (black). Each suit has 13 ranks: A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K. The J, Q and K are the face cards (12 in the deck). You will use the deck often in this course.

Two rules for a good sample space: every possible outcome is in it (it is complete), and no outcome is listed twice (the outcomes do not overlap).

Three tools for listing

Organized list. Good for small experiments. Toss two coins: S = {HH, HT, TH, TT}. Notice HT and TH are different outcomes (first coin heads, second tails is not the same as the reverse).

Table (grid). Best for two-step experiments. Rolling two dice gives a 6 × 6 grid of ordered pairs (first die, second die): 36 outcomes.

Tree diagram. Best for three or more steps. Each stage branches into its possible results; each path from left to right is one outcome.

When each stage has the same number of results no matter what happened before, you can count the outcomes by multiplying: 3 coins give 2 × 2 × 2 = 8 outcomes. (This is the fundamental counting principle, which the unit Permutations and Combinations develops in full.)

Discrete or continuous?

A sample space is discrete if you can count its outcomes, listing them one by one. Examples:

• the result of a die roll: {1, 2, 3, 4, 5, 6} (finite)
• the number of goals in a hockey game: {0, 1, 2, 3, …}
• the number of tosses needed to get the first head: {1, 2, 3, …} (it never ends, but you can still count through it)

A sample space is continuous if its outcomes are measurements that can take any value in an interval. Examples:

• the time to run 100 m (any value from about 9.5 s to, say, 30 s)
• the mass of a bag of apples, the height of a seedling, the waiting time for a bus

In a continuous sample space there are so many possible values that the probability of hitting one exact value (a run of exactly 13.000… s) is 0. Instead we ask for the probability of an interval, such as "between 13 and 14 seconds". This whole unit works with discrete sample spaces; continuous ones are the subject of the unit Continuous Distributions and the Normal Distribution.