Chapter 13.1 — Introduction to Probability

Random Experiments, Sample Space, and Events

• Random experiment — a repeatable process whose exact outcome can't be predicted beforehand, though every outcome it can possibly produce is already known. Tossing a coin and rolling a die both qualify; each toss or roll could land differently, but the full list of possibilities never changes.
• Sample space (S) — the complete list of every possible outcome a random experiment can produce. A single coin toss has sample space {head, tail}; a single die roll has sample space {1, 2, 3, 4, 5, 6}.
• Event (E) — any subset of the sample space: one particular outcome, or a group of outcomes sharing some property. "Getting an even number" on a die roll is the event {2, 4, 6}, a subset of the full six-outcome sample space.
• Elementary event — an event containing exactly one single outcome from the sample space, the smallest possible building block an event can be made from.

Every larger event is really just a bundle of elementary events stitched together. A die roll has six elementary events — {1}, {2}, {3}, {4}, {5}, {6} — each with probability 1/6 on a fair die, and any larger event's probability is simply the sum of the elementary-event probabilities it contains: "even number" bundles {2}, {4}, and {6} together, giving 1/6 + 1/6 + 1/6 = 3/6 = 1/2. Adding up the probabilities of all six elementary events in the sample space always gives exactly 1, since between them they account for every possible outcome with nothing left over and nothing double-counted.

Equally Likely and Mutually Exclusive Events

Two events are equally likely only when each one has genuinely the same chance of happening — not merely because they sound like comparable categories. Rolling a die and comparing "prime number" against "composite number" makes this distinction concrete: primes on a standard die are {2, 3, 5}, three outcomes, while composites are only {4, 6}, two outcomes. These two events are not equally likely, even though both sound like natural, parallel categories to compare.

Two events are mutually exclusive when they can never both happen from the same single trial — one occurring automatically rules the other out. A coin landing heads and that same toss landing tails are mutually exclusive by definition; so are a die showing an even number and that same roll showing an odd number, since no number is both. But a die showing an even number and that same roll showing a prime number are not mutually exclusive — the outcome 2 belongs to both events at once, so they overlap.

A standard 52-card deck supplies the same contrast. Drawing an ace and drawing a king from a single card are mutually exclusive, since no card is both ranks simultaneously. Drawing a heart and drawing a king are not — the king of hearts is a single card that satisfies both events together, so "heart" and "king" overlap rather than exclude each other.

From Experiment to Theory: Two Ways to Measure Probability

Probability can be estimated two different ways, and it's worth keeping the two genuinely separate. Experimental probability comes from actually repeating a trial many times and recording how often an outcome shows up — it's an observed frequency, not a calculation. Tossing one coin repeatedly and tracking the fraction of heads illustrates the pattern:

As the number of tosses climbs, the relative frequency keeps drifting closer to 0.5 without ever needing to land on it exactly at any particular row — that steady convergence toward a fixed value as trials increase is what experimental probability is actually built on. Theoretical (classical) probability skips the repeated trials entirely and instead reasons directly from the sample space, assuming every outcome is equally likely:

A coin toss gives P(head) = 1/2 immediately, since exactly one of the two equally likely outcomes is "head." A die roll gives P(even number) = 3/6 = 1/2, since three of the six equally likely faces — 2, 4, and 6 — satisfy the event. Neither calculation requires a single actual toss or roll; both follow directly from counting the sample space.

Complementary Events

Every event E has a complement, written E̅ or "not E" — the event consisting of every outcome in the sample space that E itself does not include. Since E and its complement together account for the entire sample space with no overlap between them, their probabilities always add to exactly 1:

Rolling a die and taking E as "prime number" gives P(E) = 3/6 = 1/2 (outcomes 2, 3, 5), and the complement "not prime" covers the remaining {1, 4, 6}, giving P(not E) = 3/6 = 1/2 as well — the two happen to split evenly here, though that's specific to this particular event, not a general rule. Taking E instead as "number greater than 4" gives P(E) = 2/6 = 1/3 (outcomes 5, 6), and its complement "number 4 or less" gives P(not E) = 4/6 = 2/3. Checking: 1/3 + 2/3 = 1, confirming the relationship holds regardless of how unevenly the split falls.

Two further boundary cases round out the picture. A sure (certain) event — one that is guaranteed to happen, such as "rolling a number less than 7" on a standard die — has probability exactly 1. An impossible event — one that can never happen, such as "rolling a 7" on that same die — has probability exactly 0. Every probability calculated throughout this chapter's exercises will fall somewhere inside this same fixed range, 0 ≤ P(E) ≤ 1, and the probabilities of every elementary event in a sample space always sum to exactly 1 across the whole space.

Where This Chapter Goes

Exercise 13.1 works entirely with the complementary-event relationship just introduced, moving between an event's probability and its complement's probability in both directions. Exercise 13.2 applies the classical probability formula directly to a long run of genuinely different random experiments — coins, dice, cards, marbles, discs, and more — each one needing nothing beyond correctly counting n(E) and n(S) for its own sample space.

Every method introduced here traces back to the same one idea: count how many outcomes satisfy what's being asked, count how many outcomes exist in total, and divide. The two exercises ahead differ only in how much work correctly identifying those two counts takes for each new scenario — the division itself never changes.