2.1 Finite Probability Spaces

Finite Probability Spaces

Let $S$ be a finite non-empty sample space, say $S = {a_{1}, a_{2}, \dots, a_{n}}$

A finite probability space or probability model, is obtained by assigning to each element $a_{i}$ in $S$ a real number, called the probability of $a_{i}$ , satisfying:

Each $p_{i}$ is non-negative, i.e. $p_{i} \geq 0$
The sum of $p_{i}$ ’s is 1, i.e. $p_{1} + p_{2} + \dots + p_{n} = 1$
The probability of an even $A$ , written $P (A)$ , is the sum of the probabilities of the elements in $A$

The singleton set ${a_{i}}$ is called an elementary event, and for notational convenience we write $P (a_{i})$ for $P ({a_{i}})$

Example

Experiment: A loaded die is tossed
Sample Space: $S = {1, 2, 3, 4, 5, 6}$
The following assignment defines a probability space:
$P (1) = \frac{3}{8}, P (2) = P (3) = P (4) = P (5) = P (6) = \frac{1}{8}$
Let $A$ be the event that an odd number appears, and $B$ the event that an even number appears

$A = {1, 3, 5}$
$B = {2, 4, 6}$

$P (A) = P (1) + P (3) + P (5) = \frac{3}{8} + \frac{1}{8} + \frac{1}{8} = \frac{5}{8}$
$P (B) = P (2) + P (4) + P (6) = \frac{1}{8} + \frac{1}{8} + \frac{1}{8} = \frac{3}{8}$