1) Probability is a numerical index between 0 and 1 that represents the likelihood of an event occurring. It is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. 2) Examples are provided to demonstrate calculating probability for events like drawing balls from a bag, rolling dice, tossing coins, and other scenarios involving chance. 3) Key terms used in probability such as sample space, outcomes, events, and how to define them are explained.

1. PROBABILITY
STD: X
MATHS- I
Maharashtra State Board of Education

2. PROBABILITY
We use certain terms such as LIKELIHOOD, CHANCES or
PROBABILITY to decide the level of UNCERTAINITY about
some results, events or social & political issues.
For example:
There is 70% chance of train no: xxx to arrive on time
There is 80% chance that I will score 90% or above in I unit test.
There is 30 % chance that the party xxx will win the
election…………..

3. PROBABILITY
• WHAT IS PROBABILITY?
Probability is a numerical index of the chances
that a certain event will occur.
like: What is the probability that the teacher’s day
will fall on Saturday in next 10 years?
PROBABILITY LIES BETWEEN 0 AND 1
CHANCES PROBABILITY
No Tr’s day on Saturday 0
10
= 0
2 tr’s day are on Saturday 2
10
=
1
5
5 tr’s day are on Saturday 5
10
=
1
2
10 tr’s day are on Saturday 10
10
= 1

4. PROBABILITY
Lets learn various TERMINOLOGY related to this unit
TERMINOLOGY EXPLANATION EXAMPLE
EXPERIEMENT
A pre planned process for the
sack of producing data
Rolling a six sided die
An Outcome
The result of single trial of an
expt.
1 or 2 or 3 or 4 or 5 or 6
Sample Space
All possible outcomes taken
together
S = {1,2,3,4,5,6}
An event
Collection of one or more
outcomes of an expt.
Specifying a certain rule
as odd no.
1 or 3 or 5

5. PROBABILITY-OUTCOMES
Now let find out how many possibilities are there
in the following.
When a coin is tossed – 2 (head or tail)
When a die is rolled - 6 (1 or 2 or 3 or4 or 5 or 6)
Any day of the week is selected randomly -7
Select an even number from 10 to 21 – 6

6. PROBABILITY SAMPLE SPACE
WHEN TWO COINS ARE TOSSED
SIMULTANEOUSLY
COIN 1 COIN 2
OUTCOMES
HEAD HEAD
HEAD TAIL
TAIL HEAD
TAIL TAIL
SAMPLE SPACE = {HH,HT,TH,TT} n(S) =4
WHEN A COIN IS TOSSED
COIN 1
OUTCOME HEAD
TAIL
SAMPLE SPACE = S= {H,T}
n(S) = 2

7. PROBABILITY SAMPLE SPACE
WHEN THREE COINS ARE TOSSED
SIMULTANEOUSLY
COIN1 COIN 2 COIN 3
OUTCOMES H H T
H T H
T H T
T T H
H T T
T H H
H H H
T T T
SAMPLE SPACE {HHT,HTH,THT,TTH,HTT,THH,HHH,TTT} n(S) =8

8. PROBABILITY SAMPLE SPACE
WHEN A DIE IS
ROLLED
DIE 1
OUTC
OMES
1,2,3,4,5,6
SAMPLE SPACE
{1,2,3,4,5,6} n(S) = 6
WHEN TWO DICE ARE ROLLED SIMULTANEOUSLY
OUTCOMES (1,1), (1,2) ,(1,3), (1,4),(1,5),(1,6)
(2,1), (2,2) ,(2,3), (2,4),(2,5),(2,6)
(3,1), (3,2) ,(3,3), (3,4),(3,5),(3,6)
(4,1), (4,2) ,(4,3), (4,4),(4,5),(4,6)
(5,1), (5,2) ,(5,3), (5,4),(5,5),(5,6)
(6,1), (6,2) ,(6,3), (6,4),(6,5),(6,6)
SAMPLE SPACE ={(1,1), (1,2) ,(1,3), (1,4),(1,5),(1,6),
(2,1), (2,2) ,(2,3), (2,4),(2,5),(2,6), (3,1), (3,2) ,(3,3),
(3,4),(3,5),(3,6), (4,1), (4,2) ,(4,3), (4,4),(4,5),(4,6),
(5,1), (5,2) ,(5,3), (5,4),(5,5),(5,6), (6,1), (6,2) ,(6,3),
(6,4),(6,5),(6,6)} } , n(S) = 36

9. PROBABILITY SAMPLE SPACE
WHEN A COIN AND A DIE ARE
THROWN SIMULTANEOUSLY
OUTCOMES COIN DIE
H 1
H 2
H 3
H 4
H 5
H 6
T 1
T 2
T 3
T 4
T 5
T 6
SAMPLE SPACE = S
={H1,H2,H3,H4,H5,H6,
T1,T2,T3,T4,T5,T6},
n(S) = 12

10. PROBABILITY SAMPLE SPACE
TWO DIGIT NUMBERS ARE FORMED USING 2,3, AND 5
WITHOUT REPEATING THE DIGITS
OUTCOMES 2 3
2 5
3 2
3 5
5 2
5 3
SAMPLE SPACE=S={23,25,32,35,52,53} n(S) = 6

11. PROBABILITY-EVENTS
A set of outcomes satisfying particular conditions is
called an event.
WHEN TWO COINS ARE
TOSSED SIMULTANEOUSLY
COIN 1 COIN 2
OUTCO
MES
HEAD HEAD
HEAD TAIL
TAIL HEAD
TAIL TAIL
SAMPLE SPACE =
{HH,HT,TH,TT} n(S) =4
WHEN TWO COINS ARE TOSSED
EVENT A-TO GET
ATLEAST ONE TAIL
S={HH,HT,TH,TT}
n(S)=4
A= {HT,TH,TT)
n(A) = 3
EVENT B- TO GET
ONLY ONE HEAD
S={HH,HT,TH,TT}
n(S)=4
B={HT,TH}
n(B) = 2
EVENT C- TO GET AT
THE MOST ONE TAIL
S={HH,HT,TH,TT}
n(S)=4
C= {HH,HT,TH}
n(C) =3
EVENT D- TO GET
NO HEAD
S={HH,HT,TH,TT}
n(S)=4
D= {TT}
n(D)=1

12. PROBABILITY-EVENTS
A BAG CONTAINS CARDS BEARING ONLY ONE NUMBER FROM 1 TO
50. ONE CARD IS DRAWN AT RANDOM FROM BAG
SAMPLE=
S{1,2,3,…….50}
n(S) = 50
EVENT A= NO. ON A
CARD DIVISIBLE BY 10
A={10,20,30,40,50}
n(A) = 5
S{1,2,3,…….50}
n(S) = 50
EVENT B = NO. ON A
CARD BEARING ONLY
ONE DIGIT
B={1,2,3,…9} n(B) = 9
S{1,2,3,…….50}
n(S) = 50
EVENT C: SUM OF THE
DIGITS IS 12
C ={39,48}, n( C) =2

13. PROBABILITY-EVENTS
TWO DIGITS NUMBER USING
DIGITS 0,1,2,3,4,5 WITHOUT
REPETITION OF THE DIGITS
OUTCOME TEN’S
PLACE
UNIT
PLACE
1 0
1 2
1 3
1 4
1 5
2 0
2 1
2 3
2 4
2 5
IN GENERAL, TENS PLACE 1 -5 NOS.
TENS PLACE 2 -5 NUMBERS,
TENS PLACE 3 -5 NUMBERS,
TENS PLACE 4 -5 NUMBERS,
TENS PLACE 5 -5 NUMBERS
SAMPLE SPACE= S=
{10,12,13,14,15,20,21,23,24,25,30,31,32,34,35,40,41,42,43,45,50,
51,52,53,54} n(S) =25
EVENT A= THE NUMBER
FORMED IS EVEN
A={10,12,14,20,24,30,32,34,
40,42,50,52,54}, n(A) =13
EVENT B = THE NUMBER
FORMED IS DIVISIBLE BY 3
B={12.15.21.24.30.42.45.51.
54}, n(S) = 9
EVENT C = THE NUMBER
FORMED IS GREATER THAN 50
C={51,52,53,54}, n(S) = 4

14. PROBABILITY-EVENTS
A COMMITTEE OF 2 MEMBERS IS TO FORMED FROM 3 BOYS AND 2 GIRLS
SAMPLE= S={B1G1,B1G2,
B2G1,B2G2,B3G1,B3G2,
B1B2,B1B3,B2B3,G1G2}
n(S) = 10
EVENT A= AT LEAST ONE GIRL
IN THE COMMITTEE
A={ B1G1,B1G2,
B2G1,B2G2,B3G1,B3G2,
G1G2} n(A)=7
EVENT B= ONE BOY AND ONE
GIRL IN A COMMITTEE
B={ B1G1,B1G2,
B2G1,B2G2,B3G1,B3G2}
n(B)=6
EVENT C= ONLY BOYS IN THE
COMMITTEE
C= {B1B2,B1B3,B2B3} n(C) =3
EVENT D= AT THE MOST ONE
GIRL IN A COMMITTEE
D={B1G1,B1G2,
B2G1,B2G2,B3G1,B3G2,
B1B2,B1B3,B2B3} n(D)=9

15. PROBABILITY-EVENTS
(1,4),(1,5),(1,6), (2,1),
(2,2) ,(2,3),
(2,4),(2,5),(2,6), (3,1),
(3,2) ,(3,3),
(3,4),(3,5),(3,6), (4,1),
(4,2) , (4,3),(4,4),(4,5),
(4,6), (5,1), (5,2) ,(5,3),
(5,4),(5,5),(5,6), (6,1),
(6,2) ,(6,3), (6,4),
(6,5),(6,6)}
N(S) = 36
the upper surface is 8 (6,2)}, n(S) = 5
Event B= sum of the digits on
the upper surface is a prime no.
B = {(1,1),(1,2),(1,4),(1,6),
(2,1), (2,3), (2,5), (3,2), (3,4),
(4,1), (4,3), (5,2), (5,6), (6,1),
(6,5)} , n(S) = 15
Event C= sum of the digits on
upper surface is multiple of 5
C={(1,4),(2,3),(3,2),(4,1),(4,6),
(5,5),(6,4)}
n(S) = 7
Event D = digit of the first die is
greater than of second die,
D={(2,1), (3,1), (3,2), (3,1),
(3,2), (4,1), (4,2), (4,3),(5,1),
(5,2) ,(5,3), (5,4), (6,1), (6,2)
,(6,3), (6,4), (6,5)} , n(S) = 17

16. PROBABILITY
When possibility (outcomes) of an event is
expressed in number, that is called
“PROBABILITY”
It is expressed as P(A) =
𝑛(𝐴)
𝑛(𝑆)
Example: no. of holidays during the month of June.
sample: (total no. of days in June) , n(s) = 30.
If no. of holidays = 6, n(A) = 6, P(A) =
6
30
=
1
5

17. PROBABILITY
A bag contains 5 balls, of which 3 are white and
2 are blue. Select any one ball from the bag
outcome W1,w2,w3,b1,b2
Sample S =
{ W1,w2,w3,
b1,b2}
n(S) = 5
EVENT A =
WHITE
BALLS
A={W1,w2,w3}
n(A) = 3
P(A) =
3
5
=
3
5
x100
= 60%
EVENT A =
BLUE BALLS
B={ b1,b2}
n(B) = 2
P(B) =
2
5
=
2
5
x100
= 40%

18. PROBABILITY
WHEN A COIN IS TOSSED
OUTCOME HEAD , TAIL
SAMPLE
S = {H, T},
n(S) =2
EVENT A =
GETTING
HEAD
A={ H},
n(A) = 1
P(A) =
1
2
=
1
2
x100
= 50%
EVENT B =
GETTING
TAIL
B={ T}
n(B) = 1
P(B) =
1
2
=
1
2
x100
= 50%

19. PROBABILITY
WHEN A DIE IS ROLLED
SAMPLE S =
{1,2,3,4,5,6},
n(S) = 6
EVENT A = NO.
ON UPPER
FACE IS
SMALLER
THAN 4
A ={1,2,3},
n(A) =3
P(A) =
3
6
=
3
6
x100
= 50%
EVENT B = NO.
ON UPPER
FACE IS
MULTIPLE OF 5
B ={5}, n(B) =1 P(B) =
1
6
=
1
6
x100
= 16.8%

20. PROBABILITY
WHEN TWO COINS ARE TOSSED,
SAMPLE S =
S={HH,HT,TH,TT}
n(S)=4
EVENT A =
GETTING AT
LEAST ONE
HEAD
A
={HH,HT,TH},
n(A) = 3
P(A) =P(A) =
3
4
=
3
4
x100 =
75%
EVENT B =
GETTING NO
HEAD
B={TT} n(B) =
1
P(B) =P(A) =
1
4
=
1
4
x100 =
25%

21. PROBABILITY
WHEN TWO DICE ARE ROLLED SIMULTANEOUSLY
S = {(1,1), (1,2) ,(1,3),
(1,4),(1,5),(1,6) (2,1), (2,2)
,(2,3), (2,4),(2,5),(2,6),
(3,1), (3,2) ,(3,3),
(3,4),(3,5),(3,6), (4,1), (4,2)
,(4,3), (4,4),(4,5),(4,6)
,(5,1), (5,2) ,(5,3),
(5,4),(5,5),(5,6), (6,1), (6,2)
,(6,3), (6,4),(6,5),(6,6)}
EVENT A = sum of
the digits on the
upper surface is at
least 10
A={(4,6),(5,5),(5,6),)
, (6,4),(6,5),(6,6)},
n(S) =6
P(A) =
6
36
=
1
6
x100 =
16.67%
Event B= sum of the
digits on the upper
surface is 33
B= {}, n(B) = 0 P(B) =
0
36
= 0 x100
= 0%
Event D = digit of
the first die is
greater than of
second die
D= {(2,1), (3,1),
(3,2) (4,1), (4,2) ,
(4,3), ,(5,1), (5,2) ,
(5,3), (5,4), (6,1),
(6,2) ,(6,3), (6,4),
(6,5)}, n(D) = 15
P(D) =
15
36
=
5
12
x100
= 41.67%

22. PROBABILITY
THERE ARE 15 TICKETS IN A BOX, NUMBERED FROM 1 TO 15. ONE
TICKET IS DRAWN AT RANDOM FROM THE BOX
SAMPLE =S=
{1,2,3,4,5,6,7,8
,9,10,11,12,13,
14,15}
EVENT A =
SHOWS AN
EVEN NO.
A
={2,4,6,8,10,12,
14}, n(S) = 7
P(A) =
7
15
=
7
15
x100
= 46.67%
EVENT B =
SHOWS NO.
MULTIPLE OF 5
B= {5,10,15}
N(B) = 3
P(B) =
3
15
=
1
5
x100
= 20%

23. PROBABILITY
TWO DIGIT NUMBER FORMED WITH 2,3,5,7,9 WITHOUT REPEATING
THE DIGITS
SAMPLE
S ={23, 25,27,
29,32,35,37,39,
52,53,57,59,72,
73,75,79,92,93,
95,97}
n (S) =20
EVENT A = AN
ODD NUMBER
A={23, 25,27,
29, 35,37,39,
53,57,59, 73,75,
79, 93,95,97},
n(s) =16
P(A) =
16
20
=
16
20
x100
= 80%
EVENT B =
MULTIPLE OF 5
B={25,35,75,95}
N(S) =4
P(B) =
4
20
=
1
5
x100
= 20%

24. REVISION
Lets learn various TERMINOLOGY related to this unit
TERMINOLOGY EXPLANATION EXAMPLE
EXPERIEMENT
A pre planned process for the
sack of producing data
Rolling a six sided die
An Outcome
The result of single trial of an
expt.
1 or 2 or 3 or 4 or 5 or 6
Sample Space
All possible outcomes taken
together
S = {1,2,3,4,5,6}
An event
Collection of one or more
outcomes of an expt.
Specifying a certain rule
as odd no.
1 or 3 or 5

25. REVISION
When possibility (outcomes) of an event is
expressed in number, that is called
“PROBABILITY”
It is expressed as P(A) =
𝑛(𝐴)
𝑛(𝑆)
Example: no. of holidays during the month of June.
sample: (total no. of days in June) , n(s) = 30.
If no. of holidays = 6, n(A) = 6, P(A) =
6
30
=
1
5