Probability — Basics and Formulas

Probability tells you how likely an event is. In basic problems, it is usually written on a scale from $0$ to $1$, where $0$ means impossible and $1$ means certain.

When outcomes are equally likely, the basic probability formula is:

$$P(A) = \frac{\text{favorable outcomes}}{\text{total outcomes}}$$

That condition matters. This ratio works for cases like a fair die or a well-shuffled deck. It does not automatically work when some outcomes are more likely than others.

Probability Definition: Outcomes And Events

An outcome is one possible result. An event is a set of outcomes you care about.

For example, when you roll a fair die, getting a $4$ is one outcome. Getting an even number is an event because it includes $2$, $4$, and $6$.

If the die is fair, the probability of rolling an even number is:

$$P(\text{even}) = \frac{3}{6} = \frac{1}{2}$$

That means the event happens half the time in the ideal fair-die model. Probability is a precise way to describe uncertainty, not just a formula to memorize.

Basic Probability Formulas To Know

Basic Formula For Equally Likely Outcomes

Use

$$P(A) = \frac{\text{favorable outcomes}}{\text{total outcomes}}$$

only when each outcome is equally likely.

Complement Rule

Sometimes it is easier to find the chance that an event does not happen:

$$P(A^c) = 1 - P(A)$$

This is especially useful for phrases like "at least one" or "not."

Addition Rule

To find the probability that $A$ or $B$ happens, use:

$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$

You subtract the overlap because outcomes in both events would otherwise be counted twice.

If the events are mutually exclusive, then $P(A \cap B) = 0$, so the rule becomes:

$$P(A \cup B) = P(A) + P(B)$$

Multiplication Rule

For independent events:

$$P(A \cap B) = P(A)P(B)$$

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

$$P(A \cap B) = P(A)P(B \mid A)$$

The condition is the important part. Do not multiply blindly unless independence is justified.

Worked Example: Probability Of At Least One $6$ In Two Rolls

Suppose you roll a fair die twice. What is the probability of getting at least one $6$?

This is a good place to use the complement rule. Instead of counting every case with a $6$, first find the chance of getting no $6$ at all.

On one roll:

$$P(\text{no }6) = \frac{5}{6}$$

Because the two rolls are independent, the probability of no $6$ on both rolls is:

$$P(\text{no }6\text{ on both rolls}) = \frac{5}{6}\cdot\frac{5}{6} = \frac{25}{36}$$

Now use the complement:

$$P(\text{at least one }6) = 1 - \frac{25}{36} = \frac{11}{36}$$

So the probability of getting at least one $6$ in two rolls is:

$$\frac{11}{36}$$

This example shows two key ideas at once: independence lets you multiply, and "at least one" problems are often easiest through the complement.

Common Probability Mistakes

One common mistake is using the ratio formula when outcomes are not equally likely. The formula $P(A) = \frac{\text{favorable}}{\text{total}}$ only works when each outcome has the same chance.

Another mistake is adding probabilities for events that overlap without subtracting the overlap. If one outcome belongs to both events, simple addition gives a value that is too large.

Students also confuse "and" with "or." In probability, "and" usually points to an intersection such as $A \cap B$, while "or" points to a union such as $A \cup B$.

A final mistake is multiplying events that are not independent. If one result changes the chance of the next, you need a conditional probability step.

When Probability Formulas Are Used

Probability is used anywhere people reason about uncertainty. Weather forecasts, medical testing, insurance, quality control, polling, and games all rely on it.

The exact model depends on the situation. Some problems use equally likely outcomes, while others use data, assumptions, or measured frequencies. The formulas still help, but only when their conditions match the problem.

Try A Similar Probability Problem

Try drawing one card from a standard deck and finding the probability of drawing a heart. Then change the question to "a heart or a king" and decide whether you need the addition rule.

If you want to check a similar setup after doing it yourself, try your own version in a math solver and compare the event definitions before you compare the final number.