U28 Introduction to Probability: Essential Formulas

Definition of Probability

For an experiment where all outcomes are equally likely, the probability of an event $E$ occurring is given by the ratio of the number of favourable outcomes to the total number of possible outcomes in the sample space $S$.

Where $0 \le P(E) \le 1$. $P(E) = 0$ means the event is impossible, and $P(E) = 1$ means the event is certain.

Probability of the Complement

The complement of an event $E$, denoted by $E'$, is the event that $E$ does not occur. The sum of their probabilities is always 1.

This is a very useful rule for simplifying calculations, especially when $P(E')$ is easier to find than $P(E)$.

For Mutually Exclusive Events

Two events $A$ and $B$ are mutually exclusive if they cannot occur at the same time, i.e., $A \cap B = \varnothing$. The probability that $A$ or $B$ occurs is the sum of their individual probabilities.

For Non-Mutually Exclusive Events

If events $A$ and $B$ can occur simultaneously, the general addition law applies to avoid double-counting the probability of their intersection $A \cap B$.

This is one of the most important formulas in basic probability. Remember that $P(A \cap B)$ represents the probability that both $A$ and $B$ occur.