# Probability with Two Variables

what are variables of probabilty:

In this article we shall discuss the probability with two variables. Probability is defined as possibility of incidence of a certain event when it is expressed quantitatively. A probability with two variables is a function or a mapping from a sample space into the real numbers. In further terms, two variables assign real values to outcomes of experiments and the output values of the mapping depend on the outcome of the trial.
Basics of Probability with Two Variables:

Outcome:

The final result of an experiment.

Example:

The experiment consists of throwing a die and the outcome would be someone of the six faces, F1,........, F6.

Sample space (S):

In each number of certain trials, the set of all probable outcomes is called the sample space.

Example:

When you roll a die, the sample space is:

S = {1, 2, 3, 4, 5, 6}
or
S = {1…6}

Event:

The sample space lists the possible outcomes of a random occurrence, that is, a subset of a sample space.

Example:

Tossing a coin, the possible events are heads or tails,

Types of Probability Variables:

Probability variables consist of following two types of variables,

Independent variables
Dependent variables

Independent variables:

Independent variable concerned with decisive the probability of obtaining one event or another based upon a single draw.

P (A and B) = P (A). P (B)

Example:

If equally dice are rolled at once, what is the probability that two 5s happen? A is the event that 5 shows on the first die, and B is the event that 5 shows on the second die.

P (A) = `1/ 6`
P (B) = `1/6`

Ans:           P (A and B) = P (A). P (B) = `(1/6)` `xx` `( 1/6)` = `1/36`

Dependent variables:

Events are dependent means each probable outcome is related to the other. The two events A and B, the probability of obtaining simultaneously.

P (A and B) = P (A). P (B|A)

P (A)`->` first event occur.

P (B|A)`->` Second event occur.

Example:

What is the probability of drawing two aces from a deck of playing cards?

Since there are 4 aces in a 52 deck of cards,

Ans:  `(4/52)` `xx` `(3/ 51)` = `1/221`