Event and Its Probability

Probability is the language of uncertainty. When you toss a coin, you don't know if it will land heads or tails — but you do know that each outcome is equally likely. That feeling of "how likely" is what probability measures.

The Intuition: What Does "Probability" Mean?

Imagine you roll a fair six-sided die. Before it stops, you can't say which number will show up. But you can say this: if you roll it many, many times, each face will appear roughly one-sixth of the time. That fraction — $\frac{1}{6}$ — is the probability of getting, say, a 4.

So probability is a number between 0 and 1 that tells you how often something happens in the long run. A probability of 0 means it never happens; a probability of 1 means it always happens.

The Building Blocks: Sample Space and Event

Before we can talk about probability, we need two things:

1. Sample space ($S$) — the set of all possible outcomes of an experiment. For a die roll, $S = \{1,2,3,4,5,6\}$.

2. Event ($E$) — any subset of the sample space. An event is a collection of outcomes you care about. For example, "rolling an even number" is the event $E = \{2,4,6\}$.

The Precise Definition

For a fair experiment — where every outcome in the sample space is equally likely — the probability of an event $E$ is:

This is the classical definition of probability. It works perfectly for dice, coins, cards, and any situation where symmetry makes all outcomes equally likely.

For the die example:

• $S$ has 6 outcomes.
• $E = \{2,4,6\}$ has 3 outcomes.
• So $P(\text{even number}) = \frac{3}{6} = \frac{1}{2}$.

Why This Makes Sense

The fraction $\frac{3}{6}$ isn't just a formula — it's a direct translation of the intuition. If you roll a fair die 600 times, you expect about 300 evens. The probability $\frac{1}{2}$ is the theoretical prediction of that long-run relative frequency. …