# Probability, Bayes'Theorem, Probability Distribution of a Random Variable

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## Bayes’ Theorem

If are non-empty mutually exclusive and exhaustive events

(i.e. ) of a sample space and be any event of non-zero probability then

Proof:

By law of total probabilities we know that

Also by law of multiplication of probabilities we have

by using (1)

This gives the proof of the Baye’s theorem let us now apply the result of Baye’s theorem to find probabilities.

Example:

Bag I contains red and black balls while another bag II contains red and black balls. One of the bags is selected at random and a ball is drawn from it. Find the probability that the ball is drawn from Bag II, if it is known that the ball drawn is red.

Solution:

Let and be the events of selecting Bag I and Bag II, respectively and be the event of selecting a red ball.

Then,

Also, (drawing a red ball from Bag I)

(drawing a red ball from Bag II)

Now, By Baye’s theorem

(bag selected is Bag II when it is known that red ball is drawn)

## Probability Distribution of a Random Variable

**Variables:**

In earlier section you have learnt to find probabilities of various events with certain conditions. Let us now consider the case of tossing a coin four times. The outcomes can be shown in a sample space as:

On this sample space we can talk about various number associated with each outcome.

For example, for each outcome, there is a number corresponding to number of heads we can call this number as .

Clearly

We find for each out come there corresponds values of ranging from to .

Such a variable is called a random variable.

**Definition:**

The probability distribution of a random variable is the distribution of probabilities to each value of . A probability distribution of a random variable is represented as

Where

The real numbers are the possible values of and is the probability of the random variable taking the value denoted as

Example:

Check whether the distribution given below is a probability distribution or not

Solution:

All probabilities are positive and less than .

Also,