Chapter 14.1 — Introduction to Probability

The Vocabulary Every Later Calculation Depends On

These four terms build directly on each other: a random experiment is performed once as a trial, and an event is simply whichever subset of that experiment's possible results happens to be of interest — "getting at least one tail" when two coins are tossed, for instance, is the event E = {HT, TH, TT}, three of the four possible outcomes. Two coins tossed together produce exactly four possible outcomes total — HH, HT, TH, TT — and any event is simply a chosen subset of that full list; "getting at least one tail" happens to include three of the four, but a different event drawn from the exact same experiment could just as easily include only one outcome, or even all four.

Two Roads to the Same Number: Experimental and Theoretical

Experimental probability is measured after the fact, by actually running trials and counting results; theoretical probability is calculated in advance, purely by counting how many outcomes exist and how many satisfy the event. Tossing a coin repeatedly shows the two converging toward each other as trials accumulate:

The relative frequency keeps drifting closer to 0.5 as the trial count grows, never landing exactly on it at any finite number of tosses but approaching it more and more closely — which is exactly the theoretical probability of heads, ½, calculated instantly without ever touching a real coin. The more trials an experiment is genuinely run for, the closer its experimental probability tends to land to the theoretical value; theoretical probability is really what experimental probability is converging toward, not a separate, unrelated number. This relationship is genuinely useful in situations where a theoretical count isn't even available — a biased coin, an oddly-weighted die, a spinner cut from irregular fractions — since experimental probability still works perfectly well there, simply by running enough trials and counting, even when no clean fraction of favourable-over-total outcomes can be worked out in advance.

Two Worked Examples, Same Formula

Find the probability of getting a head when a coin is tossed. All possible outcomes here: S = {H, T}, so n(S) = 2 total. Event E = "getting a head" = {H}, so n(E) = 1 in total. P(E) = n(E)/n(S) = 1/2.

Find the probability of getting an even number when a die is rolled. All possible outcomes here: S = {1, 2, 3, 4, 5, 6}, so n(S) = 6 total. Event A = "even number" = {2, 4, 6}, so n(A) = 3 in total. P(A) = 3/6 = 1/2. Both problems follow the identical two-step process: list every possible outcome to find n(S), then list only the outcomes satisfying the event to find n(E), before dividing one by the other. Both examples also quietly assume every outcome is equally likely — a fair coin has no bias toward heads or tails, and a fair die has no bias toward any particular face — since the formula n(E)/n(S) only ever gives the correct probability when every individual outcome inside S genuinely has the same chance of occurring; a loaded die would need a completely different calculation, most likely experimental rather than theoretical, since simply counting favourable outcomes over total outcomes would no longer reflect the actual likelihoods involved.

Multiple Coins Multiply the Outcomes

Tossing two identical coins together at once gives exactly four equally likely outcomes: HH, HT, TH, TT. P(two heads together) = 1/4 exactly. P(at least one head) = 3/4 exactly (every single outcome except TT satisfies this). P(no heads at all) = 1/4 exactly (only the single outcome TT). P(exactly one head) = 2/4 = 1/2 exactly (only the two outcomes HT and TH satisfy this particular condition). Tossing three coins instead doubles the outcome count again, to 8 — each extra coin doubles the total once more, since every existing outcome now splits into two, one with the new coin landing heads and one with it landing tails. This doubling pattern generalizes cleanly: n coins tossed together always produce exactly 2ⁿ equally likely outcomes in total, so five coins would already produce 32 distinct outcomes, and ten coins over a thousand — a number that grows far too quickly to list by hand past just a few coins, even though the underlying counting principle behind it stays exactly this simple the whole way through.

The Two Boundary Values Every Probability Sits Between

A sure event is one guaranteed to happen no matter what — rolling a die and getting a number less than or equal to 6, since every face already satisfies that condition. An impossible event, by direct contrast, never happens under any circumstance whatsoever — rolling a single ordinary die and somehow getting a 7. Every other genuine event in between always sits strictly between these two extreme boundary values, and across any complete set of outcomes for one experiment, the probabilities always add up to exactly 1, since something from that full set is guaranteed to happen on every single trial. This closing fact is a genuinely useful check on any completed probability calculation: after computing the probability of every possible outcome for a given experiment separately, adding them all back together should always land on exactly 1 — if it doesn't, either an outcome was double-counted, or one was left out of the list entirely, and the whole calculation needs revisiting before it can be trusted.

From Definitions to Genuine Word Problems

Every formula and definition here was demonstrated on the simplest possible experiments — a single coin, a single die. Exercise 14.1 applies the identical formulas to real, more complicated situations — marbles drawn from a bag, insurance records, even a dart thrown at a board — where counting outcomes correctly, rather than applying the formula itself, becomes the genuinely hard part of nearly every problem.