Probability Basics & Sample Spaces Cheat Sheet

Key Facts

• The probability of an event is $P(E)=\frac{\text{number of favorable outcomes}}{\text{number of total outcomes}}$ when all outcomes are equally likely.
• A probability must satisfy $0 \le P(E) \le 1$ for any event $E$.
• The sample space $S$ is the set of all possible outcomes, and the total probability of the sample space is $P(S)=1$.
• The complement of an event $E$ is everything not in $E$, so $P(\text{not }E)=1-P(E)$.
• Experimental probability is $P(E)=\frac{\text{number of times event }E\text{ occurs}}{\text{number of trials}}$.
• If two events cannot happen at the same time, then they are mutually exclusive and $P(A\text{ or }B)=P(A)+P(B)$.
• For two-stage experiments, a tree diagram helps list outcomes such as $\{HH, HT, TH, TT\}$ for flipping two coins.
• A fair six-sided die has sample space $S=\{1,2,3,4,5,6\}$, so the probability of rolling a $4$ is $\frac{1}{6}$.

Vocabulary

Probability

Probability is a number from $0$ to $1$ that describes how likely an event is to happen.

Outcome

An outcome is one possible result of a chance experiment, such as rolling a $3$ on a die.

Sample Space

A sample space is the complete set of all possible outcomes in an experiment.

Event

An event is a set of one or more outcomes from the sample space.

Complement

The complement of an event is the set of all outcomes in the sample space where the event does not happen.

Trial

A trial is one repetition of a probability experiment, such as one coin flip or one die roll.