COLLEGE LEVEL PROBABLITY

1. GROUP - 6

2. INTRODUCTION
Probability is the branch of mathematics concerning numerical
description of how likely an event is to occur, or how likely is it that a
promotion is true.
Probability = Number of favorable outcomes
Total Number of Outcome
EXAMPLE : let there be a basket of three balls: Red, Green, Blue. If
you want to pick a red ball we can calculate the probability of
picking a red ball.
Total no. of possibilities = 3
Total no. of favorable possibilities = 1
Therefore probability of getting a red ball = 1/3

3. KEY TERMS RELATED TO
PROBABILITY
 SAMPLE SPACE : Set of all possible outcomes of a random
experiment is called sample space. It is denoted by S.
 SAMPLE POINT : Each element of sample space is called
the sample point.
 EVENT : It is the set of variable outcome.
 MUTUALLY EXCLUSIVE EVENTS : Two events are said to be mutually
exclusive if there is no common element between them.
 EXHAUSTIVE EVENTS : The given events are exhausted if when I take
the elements in those events from the given sample space.

4. CODITIONAL PROBABLITY
When I throw a dice what is the probability that
outcome is given 3 that the outcome is odd?
The answer is 1/3 as the odd number are 1,3 and 5. So
we need the probability of getting the number 3 out of
the numbers 1,3,5. The chance is one out of 3 so the
probability is 1/3.
Note:
But P(3) = 1/6; if there is no condition. As the
probability of getting a single number named 3, out of
total 6 numbers are namely 1,2,3,4,5,6 is 1/6.

5. CONDITIONAL PROBABILITY
DEFINITION
Probability of event E is called the
conditional probability of F given that E has
already encountered, and its
defined by P(A|B)
Formula for Conditional Probability,
P(A|B) = P(A∩B)
P(B)
A B
A∩B
S

6. CONDITIONAL PROBABILITY
Example: In a survey in a class it was found that the probability of a student watching Youtube videos is
0.8 & the probability that a student is both topper & also watches Youtube videos is 0.792. what is the
probability that a student is a topper if he watches Youtube videos?
Solution:
Let P(E) denote the event that a person watches Youtube videos
Let F denote the event that a person is topper .
Then, P(E) = probability that a person watches YouTube videos = 0.8
P(E∩F) = probability that one is both topper & also watches Youtube videos = 0.792
& P(F|E)= the probability that a person is a topper if he watches Youtube videos.
According to Formula for conditional probability,
P(F|E) = P(F∩E)/ P(E)
= 0.792/0.8 = 0.99
∴ The probability that a person is a topper if he watches Youtube videos is 0.99

7. CONDITIONAL PROBABILITY
Properties of Conditional Probability
1) P(S|F) = 1
2) P(F|F) = 1
3) P((A∪B)|F) = P(A|F) + P(B|F) – P((A∩B)|F)
4) P(E'|F) = 1- P(E|F)
Example: Evaluate P(A ∪ B), if 2P(A) = P(B) = 5/13 & P(A|B) = 2/5
Solution: 2P(A) = P(B) = 5/13
 P(B) = 5/13 and P(A) = 1/2 x P(B) = 1/2 × 5/13 = 5/26
Now, the formula of conditional probability is P(A|B) = P( A∩B)/ P(B)
∴ 2/ 5 = P( A∩B) /P(B)
 P(A∩B) = 2/5 X P(B) = 2/5 × 5/13 = 2/13
Here, the formula is P(A ∪ B)=P(A)+P(B)−P(A∩B)
 P(A∪B)= 5/26 + 5/13 - 2/13 = (5+10−4 )/26 = 11/26

8. PROBABILITY
Multiplication Theorem on Probability
Let E and F be two events associated with a sample space of an
experiment.
Then
P(E ∩ F) = P(E) P(F|E), P(E) ≠ 0
= P(F) P(E|F), P(F) ≠ 0
If E, F and G are three events associated with a sample space, then
P(E∩F∩G) = P(E) P(F|E) P(G|E∩F)

9. INDEPENDENT EVENTS
Let E and F be two events associated with a sample space S. If the probability of
occurrence of one of them is not affected by the occurrence of the other, then we say
that the two events are independent. Thus, two events E and F will be independent, if
P(F | E) = P(F), provided P (E) ≠ 0
P(E | F) = P(E), provided P (F) ≠ 0
Using the multiplication theorem on probability, we have
P(E ∩ F) = P(E) P(F)
Three events A, B and C are said to be mutually
independent if all the following conditions hold:
P(A ∩ B) = P(A) P(B)
P(A ∩ C) = P(A) P(C)
P(B ∩ C) = P(B) P(C)
P(A ∩ B ∩ C) = P(A) P(B) P(C)

10. BAYE’S THEOREM
If E1, E2,..., En are mutually exclusive and
exhaustive events associated with a sample
space, and A is any event of non zero
probability, then
𝑃(𝐸𝑖 |𝐴) = 𝑃(𝐸𝑖 )𝑃(𝐴|𝐸𝑖 )
Σ𝑃(𝐸𝑖 )𝑃(𝐴|𝐸𝑖 )

11. BAYE’S THEOREM
Example: A bag contains 4 red and 4 black balls, another bag contains 2 red and 6 black balls. One of the
two bags is selected at random and a ball is drawn from the bag which is found to be red. Find the
probability that the ball is drawn from the first bag.
Solution:
Let E1= the event of selecting first bag.
E2= the event of selecting second bag.
A = the event of getting red ball.
Since there is equal chance of selecting first bag or selecting second bag,
P(E1)=P(E2)= 1/2
now P(A|E1)=P(Drawing a red ball from first bag )= 4/8
and P(A|E2)=P(Drawing a red ball from second bag )= 2/8 =1/4
Probability that ball is drawn from the first bag given that the ball drawn is red is P(E1|A)
𝑃(𝐸1|𝐴) = 𝑃(𝐴)𝑃(𝐵|𝐴) = 1/2 × 1/2 = 2/3
𝑃(𝐴)𝑃(𝐵|𝐴) + 𝑃(𝐴’)𝑃(𝐵|𝐴’) ( 1 2 × 1 2 + 1 2 × 1 4 )

12. THANK YOU