Probability Without Formulas

The forecast says 30% chance of rain tomorrow. But tomorrow only happens once. What is the 30% even claiming?

It claims this: out of many days with conditions like tomorrow's, about 30 in every 100 end up rainy. A probability is a fraction of many repeats. That is the whole definition, and it is the only one this course uses.

Probabilities live between 0 (never happens) and 1 (always happens). The same probability wears three outfits: the fraction 3/10, the decimal 0.3, and the percentage 30%. "Percent" literally means "per hundred," so 30% is 30/100, which is 0.3. Same number three ways.

A die makes this concrete. It has 6 faces, each equally likely, so the probability of rolling a four is 1/6. Divide it out: 1 ÷ 6 = 0.1666..., about 16.7%. Roll the die 600 times and you will see about 100 fours. About — not exactly. That word carries real meaning, and we will pin it down below.

Quantum computers speak this language natively. You run a circuit many times — each run is called a shot — and the machine counts how often each readout string appears. The game of this whole course is: predict those counts before pressing run.

Some outcomes cannot both happen on the same try. One roll of a die cannot be both a 1 and a 2. Outcomes like that are called exclusive.

For exclusive outcomes, chances add. The probability of rolling "a 1 or a 2" is 1/6 + 1/6. The fractions share a bottom number, so add the tops: 1 + 1 = 2, giving 2/6, which is 1/3, which is 0.3333... Makes sense: two of the six faces qualify.

Now add up all the outcomes. Six faces at 1/6 each: 1/6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6 = 6/6 = 1. Some face always comes up, so the full list of exclusive outcomes always totals exactly 1 — that is, exactly 100%.

This "totals to 1" check is the best error-catcher you will ever own. Every time you predict a quantum histogram in this curriculum, add your predicted probabilities. If they do not total 1, a mistake is hiding somewhere, guaranteed.

Flip two coins, one after the other. The first coin has no influence on the second — no wires, no magnets, nothing. Events like that are called independent.

List every combination in a little 2-by-2 table, first coin then second:

• heads, heads
• heads, tails
• tails, heads
• tails, tails

Four combinations, all equally likely, so each has probability 1/4.

Here is the rule hiding in that table: for independent events, chances multiply. In half of many repeats the first coin lands heads. Of those repeats, half have the second coin heads too. Half of a half: 0.5 × 0.5 = 0.25, which is 1/4. The table and the multiplication are the same fact.

Run the totals check from the last section: 0.25 + 0.25 + 0.25 + 0.25 = 1.00. It passes.

Three coins? Multiply again: 0.5 × 0.5 = 0.25, then 0.25 × 0.5 = 0.125, which is 1/8 — one eighth for each of 8 combinations. Eight, because three two-way choices give 2^3 outcomes, exactly the doubling you met in Powers, roots, and big numbers. The two chapters just shook hands.

If an outcome has probability 1/4 and you take 1,000 shots, how many times should it appear? Multiply: 0.25 × 1,000 = 250. That is the expected count: probability × number of tries. It is the single most-used calculation in this curriculum.

But run the experiment for real and you might count 261. Run again: 244. Again: 253. Real counts wobble around the expected count. The 250 is the center of the wobble, not a promise.

How big is the wobble? Statisticians compute it with a square-root formula, and the square root is the only part you need to feel: for 1,000 shots at probability 1/4, the typical wobble is about 14. So read "about 250" as: usually within 14 or so of 250, occasionally twice that far, almost never five times that far. A count of 259 is a pass. A count of 400 means something is genuinely wrong.

One consequence worth knowing: the wobble grows much more slowly than the count. Take 100 times more shots and the wobble only grows 10 times — so as a fraction of the total, results sharpen. That is why more shots make smoother histograms.

Keep the division of labor straight. The probabilities you predict must be exactly right — the simulator computes them exactly, and this course treats a wrong prediction as a defect. The counts merely wobble around those exact probabilities. Predict exactly; expect approximately.

You now own the three tools every later chapter leans on. Add chances for either-or outcomes. Multiply chances for independent events. And expect probability × shots, with a small wobble.

• Reading histograms is the daily work of Measurement. For a quick summary, see the measurement explainer.
• Where the probabilities come from is the next layer down. It is made of numbers called amplitudes, which you square to get probabilities. That story starts in the next chapter, Negative and complex numbers. There you will meet numbers behind the chances that can cancel each other, which ordinary chances can't do. The story goes on in Probability amplitudes.
• The totals-to-1 check becomes an official rule with a name in Lists, vectors, and normalization.
• The multiplication rule comes back for qubits in Multiple qubits.

Before you move on, take the two-coin circuit into the Lab. Run it a few times. Watch the four bars wobble around 250 without ever straying far. You can now predict that wobble.