Roulette wheel
Photo by Derek Lynn on Unsplash

Sample Spaces

The first step to building a mathematical model for probabilistic experiments is to have a model for the possible outcomes of the experiment. That's where the notion of a sample space comes in.

Definition. The sample space for an experiment is the set of all possible outcomes for the experiment.

We generally use the Greek letter \(\Omega\) (omega) to represent a sample space.

For the purpose of defining a sample space, we care only that the outcome is possible; we don't yet care how probable it is. If I flip a fair coin 1,000 times, it's unlikely in the extreme that I'll flip all heads, but that outcome is certainly one of the possible outcomes. (Note that any other outcome I actually get is equally improbable.)

Let's look at several examples. Our first examples are gambling examples.

Example: Flipping a coin

Flipping a coin is an experiment in the sense just described. The sample space contains two possible outcomes: flipping heads and flipping tails. The set-theoretic notation is

\[ \Omega = \{ H, T \} \]

Example: Rolling two (standard) dice

Rolling two dice is also an experiment. Each possible outcome involves rolling 1-6 on the first die and rolling 1-6 on the second die.

The sample space contains 36 possible outcomes. This is because there are six ways to roll the first die and six ways to roll the second die. (To learn more about how to count all possible outcomes for an experiment like this, see Rule of Product, which comes a little later in the course.) So an enumeration of the sample space is

\[ \Omega = \{ (1, 1), (1, 2), \dots, (2, 1), \dots, (6, 6) \} \]

We can also use set-builder notation to define the sample space more precisely:

\[ \Omega = \{ (d_1, d_2) \mid 1 \leq d_1 \leq 6 \land 1 \leq d_2 \leq 6 \}\]

(This reads "the set of all ordered pairs \((d_1, d_2)\) such that \(d_1\) is between 1 and 6 inclusive and \(d_2\) is between 1 and 6 inclusive". Here it's clear from the context that we're talking about natural numbers.)

Note that in this experiment, you have to roll both dice to get an outcome. Rolling a 3 on the first die and not rolling the second die at all is not a possible outcome for this experiment, because the experiment is to roll both dice.

Example: Spinning a roulette wheel

An American roulette wheel has numbers 1-36, along with 0 and 00. The 0 and 00 are green. Half of the remaining numbers are red and half are black. (See the photo above.)

Spinning a roulette wheel is an experiment where an outcome is the ball landing on one of the 38 numbers. Thus the sample space is

\[ \Omega = \{ 1, 2, 3, \dots, 34, 35, 36, 0, 00 \} \]

Example: Flipping a coin and rolling a die

For our final gambling example, consider an experiment where you flip a coin and then roll a standard die. Then one possible outcome for this experiment is flipping tails and rolling a 4.

The sample space is

\[ \Omega = \{ (c, d) \mid c \in \{H, T\} \land 1 \leq d \leq 6 \} \]

or even

\[ \Omega = \{H, T\} \times \{ 1, 2, 3, 4, 5, 6 \} \]

(Look up Cartesian product if this notation is unfamiliar.)

The examples above are all pretty straightforward. But not all experiments come from the casinos. Let's look at a few non-casino examples.

Example: Drive time to get to the office

Here the experiment is driving to the office, and the outcome is some measurement of the time it took to get there. We can arbitrarily pick minutes as a measurement of the drive time since this is how people typically measure drive times to the office.

The possible outcomes are any non-negative real number:

\[ \Omega = \{ m \in \mathbb{R} \mid m \geq 0 \} \]

Unless you actually work where you live, a drive time of 0 minutes is probably physically impossible, but logically it's a possible outcome in a way that -1 minutes isn't. Also, a drive time of one billion minutes isn't a physically possible commute time (we don't live that long), but again it's a logically possible outcome, so it's in the sample space.

Because we're dealing with real numbers, partial minutes are logically possible outcomes too. So 2.5, 3.14159265358979..., 11.2233 and so forth are all in the sample space.

Example: Number of hotel bookings in a five minute period

A travel retailer sells hotel bookings, and any given five minute period amounts to an experiment around the number of hotel bookings sold. The possible outcomes are just the natural numbers 0, 1, 2, ...:

\[ \Omega = \mathbb{N} \]

Example: Tomorrow's closing price of TSLA

Tomorrow's closing price for the stock ticker symbol TSLA (Tesla) is another experiment, and here the outcomes are amounts in US dollars and cents. It's a little tricky to specify dollars and cents in a way that avoids fractional cents (fractional cents aren't allowed here), but we can solve this by simply choosing to define the prices in terms of cents. Then once again our sample space is just the natural numbers:

\[ \Omega = \mathbb{N} \]

By now you should have a pretty good feel for sample spaces. This gives us a foundation for the next important concept, events.

Exercises

Exercise 1. Enumerate the elements of the sample space for an experiment where I flip three coins.

Exercise 2. I draw a card from a standard deck with two jokers. Describe the sample space.

Exercise 3. Consider an experiment where you're trying to shoot as many free throws (basketball) as possible in one minute. Does the sample space contain the outcome 3,433,291? Why or why not?