This document discusses the concept of probability. It defines probability as a measure of how likely an event is to occur. Probabilities can be described using terms like certain, likely, unlikely, and impossible. Mathematically, probabilities are often expressed as fractions, with the numerator representing the number of possible outcomes for an event and the denominator representing the total number of possible outcomes. The document provides examples to illustrate concepts like independent and conditional probabilities, as well as complementary events and the gambler's fallacy.

1. PROBABILIT
Y
PRESENTED BY:ISH,DIVESH & GAURAV
COURSE:B.SC(H) MATHS

2. What is probability?
Probability is the measure of how likely
an event or outcome is.
Different events have different
probabilities!

3. How do we describe probability?
You can describe the probability of an
event with the following terms:
 certain (the event is definitely going to
happen)
 likely (the event will probably happen,
but not definitely)
 unlikely (the event will probably not
happen, but it might)
 impossible (the event is definitely not
going to happen)

4. How do we express probabilities?
Usually, we express probabilities as fractions.
 The numerator shows the POSSIBLE number
of ways an event can occur.
 The denominator is the TOTAL number of
possible events that could occur.
Let’s look at an example!

5. What is the probability that I
will choose a red marble?
In this bag of marbles, there are:
3 red marbles
2 white marbles
1 purple marble
4 green marbles

6. Ask yourself the following questions:
1. How many red marbles are in the
bag?
2. How many marbles are in the bag
IN ALL?
3
10

7. This is denoted with ‘S’ and is a set
whose elements are all the possibilities
that can occur
A probability model has two components:
A SAMPLE SPACE and an assignment of
probabilities related with each OUTCOME.
***Each
element of S
is called an
outcome.
A probability of an outcome is a number and
has two properties:
1. The probability assigned to each outcome is
nonnegative.
2. The sum of all the probabilities equals 1.

8. Let's roll a die once.
S = {1, 2, 3, 4, 5, 6}
This is the sample space---all the possible outcomes
( )
Number of ways that can occur
Number of possibilities
E
P E =
probability an
event will occur
What is the probability you will roll an even number?
There are 3 ways to get an even number, rolling a 2, 4 or 6
There are 6 different numbers on the die.
( )
3 1
Even number
6 2
P = =

9. The word and in probability means the intersection of two
events.
What is the probability that you roll an even number and
a number greater than 3?
E = rolling an even number F = rolling a number greater than 3
( )P E F∩
How can E occur? {2, 4, 6}
How can F occur? {4, 5, 6}
{2,4,6} {4,5,6} {4,6}E F∩ = ∩ =
2 1
6 3
= =
The word or in probability means the union of two events.
What is the probability that you roll an even number or a
number greater than 3?
( )P E F∪
4 2
6 3
= =
{2,4,6} {4,5,6} {2,4,5,6}E F∪ = ∪ =

10. Independent Events10
Two events are independent if The outcome of
event A, has no effect on the outcome of event B.
Example: "It rained on Tuesday" and "My chair
broke at work“ are not at all related to each other.
When calculating the probabilities for independent
events you multiply the probabilities.

11. LETS TAKE AN EXAMPLE:
Let Event A:Today it will rain in Delhi.
Event B:Today I will reach office late.
Event C:Possibility of both events A&B
happening together.
GIVEN:P(A)=0.1,P(B)=0.02
Hence ; P(C)=P(A)*P(B)
=0.1*0.02
=0.002

12. •Conditional Probability
The conditional probability of
an event A (given B) is the
probability that an event A will
occur given that another event,
B, has occurred.

13. FORMULA
S
A B
( )
( )
( )
P A B
P A B
P B
∩
=

14. EXAMPLE
Suppose you roll a pair of dice: one RED in
colour while other is GREEN.
The probability that the sum of the numbers
on the dice = 9 is 4/36 since there are 4 of the
36 outcomes where the sum is 9: (3,6) (4,5)
(5,4) & (6,3).
What if you see that the RED die shows the
number 5, but you still haven’t seen the green
die? What are the chances then that the sum
of numbers we get after rolling pair of dice is
9 ??????

15. Let B = event “5 on Red die” when pair of dice
is rolled.
Let A=event “sum is 9”.
Using the formula below;we get the answer as
1/6.TRY IT OUT !!!!!!!!!!!!!
( )
( )
( )
P A B
P A B
P B
∩
=

16. Complementary Events
16
One event is the complement of another event if
the two events do not contain any of the same
outcomes, and together they cover the entire sample
space.

17. E This is read as "E complement"
and is the set of all elements in
the sample space that are not in E
Remembering our second property of probability,
"The sum of all the probabilities equals 1" we can
determine that:
( ) ( ) 1P E P E+ =
This is more often used in the form
( ) ( )1P E P E= −
If we know the probability of rain is 20% or 0.2
then the probability of the complement (no rain)
is 1 - 0.2 = 0.8 or 80%

18. GAMBLER’S FALLACY

19. The gambler's fallacy, also known as the Monte Carlo
fallacy , is the mistaken belief that if something happens
more frequently than normal during some period, then it
will happen less frequently in the future (presumably as a
means of balancing nature).For eg : A coin is tossed 5
times, and all the results comes as ‘ HEADS ’. So during the
6th
attempt , the person may feel that A TAIL IS DUE and
hence falls prey to the Fallacy by predicting a Tail ,not
thinking that the PROBABILITY OF OBTAINING
EITHER HEADS OR TAILS REMAINS SAME
The use of the term Monte Carlo fallacy originates from
the most famous example of this phenomenon, which
occurred in a Monte Carlo Casino on 18/8/1913.Then,a ball
fell in black 15 TIMES . Because of such an occuring,
gambler’s started betting on RED.But they fell prey
to the Fallacy as RED TURNED-UP after 26
attempts.

20. THANK YOU
FOR YOUR
ATTENTION
20