Probability

Conditional Probability
Type of Events
The Addition Rule
The Multiplication Rule
Binomial Distribution
At Least One Condition

Probability
For outcomes that are equally likely to occur:

$P = \dfrac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$

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

$p + q = 1$

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.

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

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

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

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

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

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

$\begin{align}
P(A \cup B \cup C) & = P(A) + P(B) + P(C) \\
& \qquad - \Big[ P(A \cap B) + P(A \cap C) + P(B \cap C) \Big] \\
& \qquad + P(A \cap B \cap C)
\end{align}$

$\begin{align}
P(A \text{ or } B \text{ or } C) & = P(A) + P(B) + P(C) \\
& \qquad - \Big[ P(A \text{ and } B) + P(A \text{ and } C) + P(B \text{ and } C) \Big] \\
& \qquad + P(A \text{ and } B \text{ and } C)
\end{align}$

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

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

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

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

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

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

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

If A, B and C are independent events, then

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

Binomial Distribution

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

$P(x) = {^n}C_x \, p^x \, q^{n - x}$

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

$P = 1 - Q$

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

Contents