Outcomes and events

To compute a probability you first have to lay out what can happen. That inventory of possibilities is the foundation everything else is built on.

Sample space and events

The sample space is the set of all possible outcomes of a situation. For one die it is ${1, 2, 3, 4, 5, 6}$ ; for a coin it is ${heads, tails}$ . An event is any collection of outcomes you care about: "the die shows an even number" is the event ${2, 4, 6}$ , a subset of the sample space.

Equally likely outcomes

When every outcome is equally likely (a fair die, a well-shuffled deck) probability becomes counting. The probability of an event is just the fraction of outcomes that make it happen:

P (event) = \frac{favourable outcomes}{total outcomes} .

The chance of an even roll is $\frac{3}{6} = \frac{1}{2}$ ; the chance of drawing a heart from a deck is $\frac{13}{52} = \frac{1}{4}$ . The whole problem reduces to counting the top and the bottom correctly.

Combining events

Two rules let you build bigger events. For the chance of either of two events, you add their probabilities, but if they overlap, you must subtract the double-counted overlap. For events that cannot both happen (rolling a $2$ or a $5$ ), there is no overlap and you simply add: $\frac{1}{6} + \frac{1}{6} = \frac{1}{3}$ .

Counting can be subtle

As situations grow, counting the outcomes gets harder and more interesting. How many ways can five cards be dealt, or three people be seated, or a lottery be drawn? Answering needs the tools of combinatorics: permutations when order matters, combinations when it does not. Getting the count right is often the whole battle; a famous puzzle like the birthday problem (that just 23 people make a shared birthday more likely than not) surprises us precisely because we count the possibilities badly. With the sample space mapped and the outcomes counted, we can move to how one event's probability shifts once we learn another has occurred.