Consider a first experiment of “Flipping a coin.” The sample space of this experiment, denoted S₁, would be the set

S₁ = {H, T}

where the letter H represents heads, and the letter T represents tails.

We could flip the coin twice, in which case we’d want to record the result of the first flip and the second flip. Using ordered pairs where the first element represents the first flip, and the second element represents the second flip, the associated sample space, denoted here as S₂, would be the set

S₂ = { (H, H), (H, T), (T, H), (T, T) }.

Alternatively, we could also just stick the letters side by side, in which case we would have the set

S₂ = { HH, HT, TH, TT }.

For the experiment of rolling a six-sided die, the sample space would be

S₃ = { 1, 2, 3, 4, 5, 6 }.

We could roll the die twice, recording the results in an ordered pair similarly to what we did in example 1.1 in which case we would have the set

S4 =	{	(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6),
		(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6),
		(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6),
		(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6),
		(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6),
		(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)	}.

Since sample spaces are sets, we could use set operations to construct them. One way to write the sample space for the experiment of rolling a die twice above is to use a Cartesian Product and write

S₄ = {1, 2, 3, 4, 5, 6} ✕ {1, 2, 3, 4, 5, 6}

For the experiment of drawing a card from a well-shuffled standard deck of playing cards, we could again use the Cartesian Product to describe the sample space as

S₅ = {♠, ♣, ♦, ♥} ✕ {A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K}

where the suits are enumerated in the first set, and in the second set we use A, J, Q, and K to represent the ace, jack, queen, and king respectively.

One point from this sample space would be the Ace of Spades (♠, A)

🂡.

A different sample point from this sample space would be the Three of Diamonds (♦, 3)

🃃.

Since there are 52 total cards in such a deck, the use of the Cartesian Product allows us to succinctly describe the sample space without having to list out every possibility.

When dealing with an experiment with infinitely many outcomes, we must use some sort of set builder notation or Cartesian Product to define the experiment’s sample space.

Consider the experiment of throwing a dart at a dartboard with radius 20. We can use polar coordinates to measure the dart’s position. The polar radius will be any real number in the interval [0, 20]. The polar angle will be any real number in the interval [0, 2π). As such, we can’t simply list out all points in the sample space.

We could use set builder notation like this:

S₆ = { (r, θ) : 0 ≤ r ≤ 20 ∧ 0 ≤ θ < 2π}.

We could also use the Cartesian Product like this:

S₅ = [0, 20] ✕ [0, 2π)

We can analyze experiments where the sample space is part finite (or countable) and part uncountably infinite.

Consider an experiment where for a particular arcade machine we want to record how many quarters it has received (a natural number or 0) and the time in minutes (a non-negative real number) it went out of service after being turned on.

In this case, we can use the Cartesian Product to define the sample space for this experiment as

S₇ = (ℕ ∪ {0}) ✕ (ℝ⁺ ∪ {0}).

Suppose for the arcade machine Space Blaster we record the sample point (123, 7200.3), meaning this machine has accepted 123 quarters ($30.75), and went out of service after being on for 7200.3 minutes (5 days and 18 minutes.) This equates to an income of 0.017 quarters per minute for this particular machine, or roughly 1 quarter per hour. Perhaps Space Blaster is not a very popular game.

On the other hand, suppose for the arcade machine Karate Chop we record the point (987, 11107.2). Karate Chop collected 987 quarters ($246.75), and went out of service after being on for 11107.2 minutes (7 days, 17 hours, 7 minutes, and 12 seconds.) This equates to a rate of 0.089 quarters per minute, or roughly 5 quarters every hour.

Consider the experiment of flipping a coin twice from Example 1.1 with sample space

S₂ = { HH, HT, TH, TT }.

Suppose we’re interested only in the outcome where the second flip shows heads. We can express the event, denoted E₁ here in two ways:

E₁ = “The second flip is heads”
E₁ = { HH, TH }.

We may only be interested in the outcome where both flips are tails. The point from the sample space is TT, the event is the set containing TT, which would be {TT}.

Let’s reconsider the experiment of rolling a single die twice and recording the order and result of the roll with sample space S₄.

There are a couple of ways we could describe the event E₂ where the dice rolls add up to 7. We can describe it using words.

E₂ = “The sum of the recorded numbers is 7”.

Since the sample space has a small number of points, we could just include all points satisfying the criteria.

E₂ = { (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1) }.

We could use set builder notation as well.

E₂ = { (x, y) : x + y = 7 }.

Now suppose we only care about outcomes where both numbers are prime. We could describe this event, denoted E₃, in a couple of different ways as well. We could simply list all such outcomes.

E₃ = { (2, 2), (2, 3), (2, 5), (3, 2), (3, 3), (3, 5), (5, 2), (5, 3), (5, 5) }.

Using set builder notation, we could write it this way:

E₃ = { (x, y) : x is prime ∧ y is prime }.

We could also simply use words:

E₃ = “Both rolls are prime numbers”.

Consider the experiment of drawing a card from a well-shuffled deck with sample space S₅.

We may care about very broad events, such as “any card whose value is odd,” or “any red card.” Sometimes, we may only care about hyper specific outcomes. Consider the event E₄ = “3 of clubs or queen of hearts.” We may as well just list out the only two cards of interest:

E₄ = { (♣, 3), (♥, Q) }, or
E₄ = { 🃓, 🂽 }

Consider the experiment of throwing a dart at a dartboard described in Example 1.4.

The radius of the Bullseye is 0.5 inches. As long as the dart is within 0.5 inches of the center of the dartboard, then the dart is in the Bullseye regardless of the angle. Hence, for the event E₅ of “Hitting the Bullseye,” we could write this mathematically as

E₅ = { (r, θ) : 0 ≤ r ≤ 0.5 }.

Now consider a different event E₆ = “The dart lands such that it anywhere from 10 inches to 12 inches from the center, and is at a polar angle from π/3 to π/2.” This event can be described mathematically as follows:

E₆ = { (r, θ) : 10 ≤ r ≤ 12 ∧ π/3 ≤ θ ≤ π/2}.

A typical dartboard is divided into twenty angular sections, and 6 radial sections. If you wanted to describe the event of a dart landing in a specific section, you can just list the radial and angular bounds for that specific section.

Consider the experiment from Example 1.5 where for an arcade machine, we record how many quarters it has collected since being turned on, and the time in minutes it takes for it to go out of service after being turned on.

Suppose we only care about outcomes where the machine has collected at least four quarters. We could describe this event as

E₇ = “The machine has collected at least 4 quarters”
E₇ = { (x, y) : x ≥ 4 }

We may only care about machines that collect between one and three quarters inclusive that are on for no longer than five minutes. In this case, we could write

E₈ = “The machine collected between 1 and 3 quarters inclusive, and is on for no longer than 5 minutes.”
E₈ = { (x, y) : 1 ≤ x ≤ 3 ∧ y ≤ 5 }