# Probability

**Probability**

For outcomes that are equally likely to occur:

If the probability for an event to happen is *p* and the probability for it to fail is *q*, then

## Conditional Probability

The probability that event *B* occurs given that event *A* has occurred is denoted by *P*(*B*/*A*). This is called the *conditional probability*.

$P(B/A) = \dfrac{P(A \text{ and } B)}{P(A)}$

## Type of Events

**Dependent and Independent Events**

If the probability of one event does not affect the probability of another event, then, the events are said to be *independent*, otherwise, the events are *dependent*.

**Mutually Exclusive Events**

Two or more events are said to be *mutually exclusive* if each event cannot happen in a single moment.

## The Addition Rule

This is also called the **OR Rule** from which mutually exclusive events can be calculated. The probability that events *A* or *B* will occur is given by

$P(A \text{ or } B) = P(A) + P(B) – P(A \text{ and } B)$

The probability that events *A* or *B* or *C* will occur is given by

P(A \cup B \cup C) & = P(A) + P(B) + P(C) \\

& \qquad - \Big[ P(A \cap B) + P(A \cap C) + P(B \cap C) \Big] \\

& \qquad + P(A \cap B \cap C)

\end{align}$

$\begin{align}

P(A \text{ or } B \text{ or } C) & = P(A) + P(B) + P(C) \\

& \qquad - \Big[ P(A \text{ and } B) + P(A \text{ and } C) + P(B \text{ and } C) \Big] \\

& \qquad + P(A \text{ and } B \text{ and } C)

\end{align}$

If *A*, *B* and *C* are mutually exclusive events, then

## The Multiplication Rule

This is also called the **AND Rule** from which dependent and independent events can be calculated. The probability that two events *A* and *B* will occur in sequence is

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

$P(A \text{ and } B) = P(A) \times P(B/A)$

$P(A \text{ and } B) = P(B) \times P(A/B)$

The probability that events *A* and *B* and *C* will occur is given by

If *A*, *B* and *C* are independent events, then

## Binomial Distribution

The following formula can be used for repeated independent trials having the same probability of success.

where, in a single trial, *p* is the probability of success and *q* is the probability of failure.

## At Least One Condition

The probability of an event to happen at least once is

Where *Q* is the probability of the event to totally fail.