Theoretical Probability Formula

# Theoretical Probability Formula

## Random Experiment

A Random experiment is an
experiment which satisfies the following three conditions.
a)      It must have two or more outcomes.
b)     The outcomes must be uncertain
c)      The experiment can be repeated any number of times under similar conditions.
Examples: -
a)Tossing a Coin
b) Throwing a die.

## Theoretical Probability formula

Let us consider a random experiment.
Let A be an outcome of the random experiment .  Then A is called an event.
The theoretical probability of the event A is given by
P(A) = Number of outcomes favorable to A / Total number of outcomes
Theoretical probability is also known as Classical
probability or A Priori probability.

Let us consider a few examples

Example 1: -
Consider an experiment of throwing a die.
Find the probability of getting
(i) an even number     (ii) a number 5       (iii) getting a number less than 4
Solution : -
We know that the formula for calculating probability is
P(A) = Number of outcomes favorable to A / Total number of outcomes
When a die is thrown the possible outcomes are 1, 2, 3, 4, 5, 6.
So the sample space is S = {1, 2, 3, 4, 5, 6}
Therefore total number of outcomes=6
(i)   We have to find the probability of getting an even number.
2,4,6 are the even numbers in the sample space.
So number of outcomes favorable to getting an even numbers is 3.
P(getting an even number) = 3/6 = 1/2
(ii)     We have to find the probability of getting a number 5.
There is only one 5 in the possible outcomes.  So number of outcomes favorable to getting a number 5  is 1.
Therefore P(getting a number  5) = 1/6
(iii) We have to find the probability of getting a number less than 4.
We can see that 1,2,3 are the numbers which are less than 4.
So number of outcomes favorable to getting a number less than 4 is 3.
P(getting a number less than 4) = 3/6 = 1/2

Example 2: -
Lots are numbered from 1 to 100.  They are well shuffled and a lot is drawn at
random.  What is the probability of getting
(i) a multiple of number 10
(ii) a number greater than 75
(iii) an even number ?
Solution: -
We know that the formula for calculating probability is
P(A) = Number of outcomes favorable to A / Total number of outcomes
There are 100 tickets
Total number of outcomes =100
(i)  We have to find the probability of getting a multiple of number 10.
We can see that 10,20,30,40,50,60,70,80,90 and 100 are the multiples of 10.
So the number of favorable outcomes to multiple of number 10 = 10
Therefore P(getting a multiple of number 10) = 10/100 = 0.1
(ii)  We have to find the probability of getting a number greater than 75 .
All the numbers starting from 76  and up to 100, are greater than 75. There are 25 numbers are greater than 75.
So the number of outcomes favorable to numbers greater than 75 = 25
P( getting a number greater than 75) = 25/100 = 0.25
(iii)   We have to find the probability of getting an even number.
We know that 50 of the 100 numbers will be even and 50 odd.
So the number of outcomes favorable to even numbers = 50
Therefore P( getting an even number) = 50/100 = 0.5

Example 3: -
A bag contains 3 white, 4 blue and 6 red balls.  A ball is drawn at random. Find that probability that it is
(i) white
(ii) red
(iii) not blue
(iv) blue or red
Solution: -
We know that the formula for calculating probability is
P(A) = Number of outcomes favorable to A / Total number of outcomes
There are 3 white, 4 blue and 6 red balls.
Total number of outcomes = 3 + 4 + 6 = 13
i)  We have to find the probability of getting a white ball
We can see that there are 3 white balls in the bag.
So the number of favorable outcomes for the event white ball = 3
Therefore P(getting a white ball) = 3/13
ii)  We have to find the probability of getting a red ball
We can see that there are 6 red balls in the bag.
So the number of favorable outcomes for the event red ball = 6
Therefore P(getting a red ball) = 6/13
iii)  We have to find the probability of getting ball which is not blue.
We can see that there are 3 white balls and 6 red balls in the bag.
So the number of favorable outcomes for the event not a blue ball = 9
Therefore P(getting a ball which is not blue) = 9/13
iv)  We have to find the probability of getting ball which is blue or red.
We can see that there are 4 blue balls and 6 red balls in the bag.
So the number of favorable outcomes for the event blue or red ball = 10
Therefore P(getting a ball which is blue or red) = 10/13