Probability Cheat Sheet

Formulae

Theoretical (classical) probability

P(E) = \frac{\text{number of outcomes favourable to } E}{\text{total number of possible outcomes}}

E is an event; all outcomes are assumed equally likely.

Range of probability

0 \le P(E) \le 1

The probability of any event lies between 0 and 1 inclusive.

Probability of the complementary event

P(E) + P(\bar{E}) = 1, \qquad P(\bar{E}) = 1 - P(E)

Ē (not E) is the event that E does not happen.

Sure and impossible events

P(\text{sure}) = 1, \qquad P(\text{impossible}) = 0

An event that must happen has probability 1; one that cannot happen has probability 0.

Sum of probabilities of all outcomes

\sum P(E_{i}) = 1

For the set of all elementary outcomes of an experiment.

Probability: A measure of how likely an event is to happen, expressed as a number from 0 to 1.
Random experiment: An experiment whose outcome cannot be predicted with certainty.
Outcome: A single possible result of a random experiment.
Sample space: The set of all possible outcomes of an experiment.
Event: A collection of one or more outcomes of an experiment.
Equally likely outcomes: Outcomes that have the same chance of occurring.
Complementary event: The event that a given event E does not occur; written Ē or “not E”.
Sure event: An event that is certain to occur, with probability 1.
Impossible event: An event that cannot occur, with probability 0.

The theoretical probability of an event E is the ratio of favourable outcomes to total outcomes, when all outcomes are equally likely.
A probability can never be negative or greater than 1.
The probabilities of all elementary outcomes of an experiment add up to 1.
The sum of the probability of an event and that of its complement is 1.
An impossible event has probability 0 and a sure event has probability 1.
For a fair coin there are two outcomes, for a fair die six outcomes, and for a standard deck of cards 52 outcomes.
A standard deck has 26 red and 26 black cards, and 12 face cards (king, queen, jack).
The probability of an event is used to make predictions and decisions in real-life situations.