Probability — Basics and Formulas

Probability tells you how likely an event is, on a scale from $0$ (impossible) to $1$ (certain). When outcomes are equally likely, the basic formula is

where $A$ is the event, "favorable outcomes" are the ones in $A$, and "total outcomes" is every possible result. That condition matters: this ratio works for a fair die or a well-shuffled deck, but not when some outcomes are more likely than others.

Why the Ratio Works

An outcome is one possible result; an event is a set of outcomes you care about. Rolling a fair die, getting a $4$ is one outcome, while "even" is an event containing $2$, $4$, and $6$. The ratio formula is just counting: if all six faces are equally likely, then three of them are even, so

Each favorable outcome contributes the same share of certainty, which is exactly why you can divide counts. Break equal likelihood and the division stops being valid — that single assumption is what the whole formula rests on.

The Core Formulas

Equally likely outcomes (only when each outcome has the same chance):

Complement rule — for the chance an event does not happen, handy for "at least one" and "not":

Addition rule — for "$A$ or $B$":

You subtract the overlap because outcomes in both events would otherwise be double-counted. If the events are mutually exclusive, $P(A \cap B) = 0$ and it reduces to $P(A \cup B) = P(A) + P(B)$.

Multiplication rule — for independent events:

If the second event depends on the first, use conditional probability:

Do not multiply blindly unless independence is justified.

Worked Example: At Least One $6$ in Two Rolls

Roll a fair die twice — what is the probability of at least one $6$? This is a natural place for the complement rule: instead of counting every case with a $6$, find the chance of no $6$ first.

On one roll:

The rolls are independent, so the chance of no $6$ on both is

Now take the complement:

So the answer is $\frac{11}{36}$. Two ideas show up at once: independence lets you multiply, and "at least one" problems are often easiest through the complement.

Practice and Self-Check

Draw one card from a standard deck and find the probability of a heart. Then change the question to "a heart or a king" and decide whether you need the addition rule — and if so, what overlap you must subtract. Before comparing final numbers with anyone, compare your event definitions first, since that is where most disagreements actually come from.

Calculation Traps to Avoid

• Using the ratio formula when outcomes are not equally likely. It only holds when each outcome has the same chance.
• Adding probabilities for overlapping events without subtracting the overlap, which inflates the result.
• Confusing "and" with "or": "and" usually means an intersection $A \cap B$, "or" means a union $A \cup B$.
• Multiplying events that are not independent. If one result changes the chance of the next, you need a conditional probability step.

The exact model depends on the situation — equally likely outcomes in some problems, measured frequencies or assumptions in others — but every formula here works only when its condition matches the problem.