Statistics › 🎲 Probability

What Is Probability?

The number between 0 and 1 that measures how likely something is — and what it really means in the long run.

Intro StatisticsAP Statistics level
💡
The big idea: A probability is a single number, between 0 and 1, that measures how likely something is. When outcomes are equally likely you can find it by counting — favorable outcomes divided by all outcomes. And it carries a concrete real-world meaning: repeat the experiment many times and the fraction of the time the event happens closes in on that number. That link between the probability you calculate and the proportion you observe is the foundation the rest of statistics is built on.
🎯 By the end, you'll be able to
  • Read a probability as a number on the 0-to-1 scale and say what 0, 0.5, and 1 mean
  • Compute the probability of an event by counting favorable outcomes over total equally likely outcomes
  • Explain probability as a long-run relative frequency and state the law of large numbers
  • Use the complement rule P(not A) = 1 - P(A) to find probabilities quickly
📎 You should already know
  • Fractions, decimals, and percentages
  • Counting basic outcomes of an experiment

What does 'probability' actually mean?

Ask someone what a 70% chance of rain means and you will usually get a shrug. Probability is one of those ideas everyone uses and few can pin down. At its core it is simple: a probability is a number that measures how likely something is, on a fixed scale running from impossible to certain.

An event that cannot happen has probability 0. An event that is certain has probability 1. Everything else lands somewhere in between: a fair coin landing heads sits right in the middle at 0.5.

🔑 Probability lives on a 0-to-1 scale
Every probability is a number between 0 and 1 (often written as a percentage from 0% to 100%). Closer to 0 means less likely; closer to 1 means more likely; exactly 0.5 means an even chance either way. A probability can never be negative or larger than 1 — if your arithmetic ever gives you 1.4, something has gone wrong.

Counting equally likely outcomes

When an experiment has a handful of outcomes that are all equally likely — the six faces of a fair die, the slices of an even spinner — probability becomes pure counting. You ask two questions: how many outcomes count as a 'win' (the favorable outcomes), and how many outcomes are there in total?

The probability is just the ratio of the two.

\[ P(A) = \frac{\text{number of favorable outcomes}}{\text{total number of equally likely outcomes}} \]
The equally-likely (classical) definition of probability. It applies only when every outcome truly has the same chance.

The long-run meaning

But what does \( P = 0.5 \) really mean for a single coin flip? You cannot get half a head. The honest answer is about the long run: if you flipped the coin thousands of times, the proportion of heads would be very close to 0.5.

This is the relative-frequency view of probability: the probability of an outcome is the fraction of the time it happens when you repeat the experiment over and over. The more repetitions, the closer that fraction tends to sit to the theoretical value.

The tool below lets you watch this happen. Pick an experiment — a fair coin (tracking Heads, theoretical 0.5), a fair die (tracking a roll of 6, theoretical about 0.167), or a spinner (tracking a shaded quarter, theoretical 0.25). Press Trial x1 to go one step at a time, or Run 100 to fast-forward. The jagged line is the running proportion of wins so far; the flat dashed line is the theoretical probability.

🎮 Interactive: empirical vs theoretical probability LIVE
Every press runs honest random trials. Early on the running proportion (solid line) swings wildly; as the trials pile up it settles onto the theoretical value (dashed line). Try Run 100 several times and watch the line flatten out.
✨ The law of large numbers
Notice the pattern: after just a few trials the proportion can be far off — five heads in a row is common — but after hundreds or thousands of trials it hugs the theoretical line. This is the law of large numbers: as the number of independent trials grows, the empirical proportion converges to the true probability. It is the bridge between the theoretical probability you calculate and the empirical proportion you observe. One caution: it does not mean a run of heads has to be 'balanced out' by tails — each flip has no memory. The proportion settles simply because those early swings get diluted by more and more trials.

The complement: probability of 'not'

Often the easiest way to find a probability is to work out the chance of the event not happening. Since every trial either produces the event or does not, those two probabilities must add up to 1.

The event 'A does not happen' is called the complement of A. Rearranging gives a rule that saves a lot of counting:

\[ P(\text{not } A) = 1 - P(A) \]
The complement rule: the probability an event does not occur is 1 minus the probability that it does.
📝 Worked example: A standard six-sided die is rolled once. (a) What is the probability of rolling a 5? (b) What is the probability of NOT rolling a 5?
  1. List the outcomes. A fair die has 6 equally likely faces: 1, 2, 3, 4, 5, 6.
  2. (a) Favorable outcomes for 'roll a 5' is just one face, the 5. So \( P(5) = \frac{1}{6} \approx 0.167 \).
  3. (b) Rather than counting the other five faces, use the complement rule: \( P(\text{not } 5) = 1 - P(5) = 1 - \frac{1}{6} \).
✓ P(rolling a 5) = 1/6 ≈ 0.167, and P(not rolling a 5) = 5/6 ≈ 0.833. The two add to 1, as every pair of complementary probabilities must.

Check your understanding

1. On the probability scale, a value of 0 means the event is…
0 is the bottom of the scale: the event cannot happen. Certain events sit at 1, and an even chance is 0.5.
2. A fair spinner has 5 equal slices, 2 of them shaded red. The probability of landing on red is…
Equally likely outcomes: favorable ÷ total = 2 ÷ 5 = 0.4. (A probability can never be 2.5 — it must stay between 0 and 1.)
3. You flip a fair coin 10 times and happen to get 7 heads (a proportion of 0.70). As you keep flipping, the law of large numbers predicts that…
The running proportion converges toward the true 0.5 as trials grow — but not because the coin 'remembers' or 'balances out'. Early swings simply get diluted by more flips.
4. The probability of rain tomorrow is P(rain) = 0.3. Using the complement rule, P(no rain) is…
P(not A) = 1 − P(A) = 1 − 0.3 = 0.7. The event and its complement always sum to 1.
✅ Key takeaways
  • A probability is a number from 0 (impossible) to 1 (certain); 0.5 is an even chance.
  • For equally likely outcomes, P(A) = favorable outcomes ÷ total outcomes.
  • Probability means long-run relative frequency: the fraction of the time an event happens over many repeats.
  • The law of large numbers says that empirical proportion converges to the theoretical probability as trials grow — with no 'memory' balancing things out.
  • The complement rule, P(not A) = 1 - P(A), is a fast way to find the chance an event does not occur.