Probability Class 11 Notes Maths Chapter 14

 Page 1


 
KEY POINTS
? Random Experiment: If an experiment has more than one 
possible outcome and it is not possible to predict the outcome in 
advance then experiment is called random experiment. 
? Sample Space: The collection or set of all possible outcomes of 
a random experiment is called sample space associated with it. 
Each element of the sample space(set) is called a sample point. 
? Some examples of random experiments and their sample 
spaces 
(i) A coin is tossed 
S = {H, T}, n(S) = 2 
Where n(S) is the number of elements in the sample space S. 
(ii) A die is thrown 
S = { 1, 2, 3, 4, 5, 6], n(S) = 6 
(iii) A card is drawn from a pack of 52 cards n (S) = 52. 
(iv) Two coins are tossed 
S = {HH, HT, TH, TT}, n(S) = 4. 
(v) Two dice are thrown 
Page 2


 
KEY POINTS
? Random Experiment: If an experiment has more than one 
possible outcome and it is not possible to predict the outcome in 
advance then experiment is called random experiment. 
? Sample Space: The collection or set of all possible outcomes of 
a random experiment is called sample space associated with it. 
Each element of the sample space(set) is called a sample point. 
? Some examples of random experiments and their sample 
spaces 
(i) A coin is tossed 
S = {H, T}, n(S) = 2 
Where n(S) is the number of elements in the sample space S. 
(ii) A die is thrown 
S = { 1, 2, 3, 4, 5, 6], n(S) = 6 
(iii) A card is drawn from a pack of 52 cards n (S) = 52. 
(iv) Two coins are tossed 
S = {HH, HT, TH, TT}, n(S) = 4. 
(v) Two dice are thrown 
 
(1 ,1),(1 ,2),(1 ,3),(1 ,4),(1 ,5),(1 ,6)
(2,1 ),(2,2),........................,(2,6)
( ) 36.
(6,1 ),(6,2),(6,3),(6,4),(6,5),(6,6)
S n s
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
?
?
 
(vi) Two cards are drawn from a well shuffled pack of 52cards 
(a) with replacement n(S) = 52 × 52 
(b) without replacement 
52
2
( ) n S C ?
? Event: A subset of the sample space associated with a random 
experiment is called an event. 
? Elementary or Simple Event: An event which has only one 
Sample point is called a simple event. 
? Compound Event:An event which has more than one Sample 
point is called a Compound event. 
? Sure Event: If event is same as the sample space of the 
experiment, then event is called sure event. 
? Impossible Event: Let S be the sample space of the 
experiment, ?? S, ?is called impossible event. 
? Exhaustive and Mutually Exclusive Events: Events E1, E2, 
E3……..En are such that 
(i) E1U E2UE3U…….. UEn = S then Events E1, E2, E3……..En
are called exhaustive events. 
(ii) Ei ? ? Ej = ? for every i ? j then Events E1, E2, E3……..En are 
called mutually exclusive. 
Page 3


 
KEY POINTS
? Random Experiment: If an experiment has more than one 
possible outcome and it is not possible to predict the outcome in 
advance then experiment is called random experiment. 
? Sample Space: The collection or set of all possible outcomes of 
a random experiment is called sample space associated with it. 
Each element of the sample space(set) is called a sample point. 
? Some examples of random experiments and their sample 
spaces 
(i) A coin is tossed 
S = {H, T}, n(S) = 2 
Where n(S) is the number of elements in the sample space S. 
(ii) A die is thrown 
S = { 1, 2, 3, 4, 5, 6], n(S) = 6 
(iii) A card is drawn from a pack of 52 cards n (S) = 52. 
(iv) Two coins are tossed 
S = {HH, HT, TH, TT}, n(S) = 4. 
(v) Two dice are thrown 
 
(1 ,1),(1 ,2),(1 ,3),(1 ,4),(1 ,5),(1 ,6)
(2,1 ),(2,2),........................,(2,6)
( ) 36.
(6,1 ),(6,2),(6,3),(6,4),(6,5),(6,6)
S n s
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
?
?
 
(vi) Two cards are drawn from a well shuffled pack of 52cards 
(a) with replacement n(S) = 52 × 52 
(b) without replacement 
52
2
( ) n S C ?
? Event: A subset of the sample space associated with a random 
experiment is called an event. 
? Elementary or Simple Event: An event which has only one 
Sample point is called a simple event. 
? Compound Event:An event which has more than one Sample 
point is called a Compound event. 
? Sure Event: If event is same as the sample space of the 
experiment, then event is called sure event. 
? Impossible Event: Let S be the sample space of the 
experiment, ?? S, ?is called impossible event. 
? Exhaustive and Mutually Exclusive Events: Events E1, E2, 
E3……..En are such that 
(i) E1U E2UE3U…….. UEn = S then Events E1, E2, E3……..En
are called exhaustive events. 
(ii) Ei ? ? Ej = ? for every i ? j then Events E1, E2, E3……..En are 
called mutually exclusive. 
 
Then we say that E1, E2,……..En partitions the sample space S. 
Probability of an Event:For a finite sample space S with 
equally likely outcomes, probability of an event A is defined as: 
( )
( )
( )
n A
P A
n S
?
 
where n(A) is number of elements in A 
and n(S) is number of elements in set S and 0 ? P (A) ? 1. 
? (a) If A and B are any two events then 
P(A or B) = P(A ? B) = P(A) + P(B) – P(A ? B) 
= P(A) + P(B) – P (A and B) 
(b) If A and B are mutually exclusive events, then 
P(A ? B) = P(A) + P(B) (since P(A ? B)=0 for mutually 
exclusive events) 
(c) ? ? P A P(A) 1 ? ? or P(A) + P(not A) = 1 
(d) P (Sure event) = P(S) = 1 
(e) P (impossible event) =  P ( ?)  = 0 
(f) P(A – B) = P(A) – P(A ? B) = P(A ? B ) 
(g) ( ) ( ) ( ) ( ) P B A P B P A B P B A ? ? ? ? ? ? 
(h) ( ) ( ) 1 ( ) P A B P A B P A B ? ? ? ? ? ? 
(i) ( ) ( ) 1 ( ) P A B P A B P A B ? ? ? ? ? ? 
Page 4


 
KEY POINTS
? Random Experiment: If an experiment has more than one 
possible outcome and it is not possible to predict the outcome in 
advance then experiment is called random experiment. 
? Sample Space: The collection or set of all possible outcomes of 
a random experiment is called sample space associated with it. 
Each element of the sample space(set) is called a sample point. 
? Some examples of random experiments and their sample 
spaces 
(i) A coin is tossed 
S = {H, T}, n(S) = 2 
Where n(S) is the number of elements in the sample space S. 
(ii) A die is thrown 
S = { 1, 2, 3, 4, 5, 6], n(S) = 6 
(iii) A card is drawn from a pack of 52 cards n (S) = 52. 
(iv) Two coins are tossed 
S = {HH, HT, TH, TT}, n(S) = 4. 
(v) Two dice are thrown 
 
(1 ,1),(1 ,2),(1 ,3),(1 ,4),(1 ,5),(1 ,6)
(2,1 ),(2,2),........................,(2,6)
( ) 36.
(6,1 ),(6,2),(6,3),(6,4),(6,5),(6,6)
S n s
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
?
?
 
(vi) Two cards are drawn from a well shuffled pack of 52cards 
(a) with replacement n(S) = 52 × 52 
(b) without replacement 
52
2
( ) n S C ?
? Event: A subset of the sample space associated with a random 
experiment is called an event. 
? Elementary or Simple Event: An event which has only one 
Sample point is called a simple event. 
? Compound Event:An event which has more than one Sample 
point is called a Compound event. 
? Sure Event: If event is same as the sample space of the 
experiment, then event is called sure event. 
? Impossible Event: Let S be the sample space of the 
experiment, ?? S, ?is called impossible event. 
? Exhaustive and Mutually Exclusive Events: Events E1, E2, 
E3……..En are such that 
(i) E1U E2UE3U…….. UEn = S then Events E1, E2, E3……..En
are called exhaustive events. 
(ii) Ei ? ? Ej = ? for every i ? j then Events E1, E2, E3……..En are 
called mutually exclusive. 
 
Then we say that E1, E2,……..En partitions the sample space S. 
Probability of an Event:For a finite sample space S with 
equally likely outcomes, probability of an event A is defined as: 
( )
( )
( )
n A
P A
n S
?
 
where n(A) is number of elements in A 
and n(S) is number of elements in set S and 0 ? P (A) ? 1. 
? (a) If A and B are any two events then 
P(A or B) = P(A ? B) = P(A) + P(B) – P(A ? B) 
= P(A) + P(B) – P (A and B) 
(b) If A and B are mutually exclusive events, then 
P(A ? B) = P(A) + P(B) (since P(A ? B)=0 for mutually 
exclusive events) 
(c) ? ? P A P(A) 1 ? ? or P(A) + P(not A) = 1 
(d) P (Sure event) = P(S) = 1 
(e) P (impossible event) =  P ( ?)  = 0 
(f) P(A – B) = P(A) – P(A ? B) = P(A ? B ) 
(g) ( ) ( ) ( ) ( ) P B A P B P A B P B A ? ? ? ? ? ? 
(h) ( ) ( ) 1 ( ) P A B P A B P A B ? ? ? ? ? ? 
(i) ( ) ( ) 1 ( ) P A B P A B P A B ? ? ? ? ? ? 
 
? Addition theorem for three events 
Let A, Band Cbe any three events associated with a random 
experiment, then 
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ).
P A B C P A P B P C P A B P B C P C A
P A B C
? ? ? ? ? ? ? ? ? ? ?
? ? ?
? Axiomatic Approach to Probability: 
Let S be a sample space containing elementary outcomes 
1 2
, ,...........
n
w w w
i.e. ? ?
1 2
, ,...........
n
S w w w ?
(i) 0 ( ) 1
i
P w ? ? for each 
i
w S ?
(ii) 
1 2
( ) ( ) ........... ( ) 1
n
P w P w P w ? ? ? ? 
(iii) ( ) ( )
i
P A P w ? ? for any event A containing elementary
events 
i
w . 
Verbal 
description of 
the event 
Equivalent Set 
Theoretic 
Notation 
Venn Diagram 
Not A 
' A or A 
A or B A B ?
Page 5


 
KEY POINTS
? Random Experiment: If an experiment has more than one 
possible outcome and it is not possible to predict the outcome in 
advance then experiment is called random experiment. 
? Sample Space: The collection or set of all possible outcomes of 
a random experiment is called sample space associated with it. 
Each element of the sample space(set) is called a sample point. 
? Some examples of random experiments and their sample 
spaces 
(i) A coin is tossed 
S = {H, T}, n(S) = 2 
Where n(S) is the number of elements in the sample space S. 
(ii) A die is thrown 
S = { 1, 2, 3, 4, 5, 6], n(S) = 6 
(iii) A card is drawn from a pack of 52 cards n (S) = 52. 
(iv) Two coins are tossed 
S = {HH, HT, TH, TT}, n(S) = 4. 
(v) Two dice are thrown 
 
(1 ,1),(1 ,2),(1 ,3),(1 ,4),(1 ,5),(1 ,6)
(2,1 ),(2,2),........................,(2,6)
( ) 36.
(6,1 ),(6,2),(6,3),(6,4),(6,5),(6,6)
S n s
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
?
?
 
(vi) Two cards are drawn from a well shuffled pack of 52cards 
(a) with replacement n(S) = 52 × 52 
(b) without replacement 
52
2
( ) n S C ?
? Event: A subset of the sample space associated with a random 
experiment is called an event. 
? Elementary or Simple Event: An event which has only one 
Sample point is called a simple event. 
? Compound Event:An event which has more than one Sample 
point is called a Compound event. 
? Sure Event: If event is same as the sample space of the 
experiment, then event is called sure event. 
? Impossible Event: Let S be the sample space of the 
experiment, ?? S, ?is called impossible event. 
? Exhaustive and Mutually Exclusive Events: Events E1, E2, 
E3……..En are such that 
(i) E1U E2UE3U…….. UEn = S then Events E1, E2, E3……..En
are called exhaustive events. 
(ii) Ei ? ? Ej = ? for every i ? j then Events E1, E2, E3……..En are 
called mutually exclusive. 
 
Then we say that E1, E2,……..En partitions the sample space S. 
Probability of an Event:For a finite sample space S with 
equally likely outcomes, probability of an event A is defined as: 
( )
( )
( )
n A
P A
n S
?
 
where n(A) is number of elements in A 
and n(S) is number of elements in set S and 0 ? P (A) ? 1. 
? (a) If A and B are any two events then 
P(A or B) = P(A ? B) = P(A) + P(B) – P(A ? B) 
= P(A) + P(B) – P (A and B) 
(b) If A and B are mutually exclusive events, then 
P(A ? B) = P(A) + P(B) (since P(A ? B)=0 for mutually 
exclusive events) 
(c) ? ? P A P(A) 1 ? ? or P(A) + P(not A) = 1 
(d) P (Sure event) = P(S) = 1 
(e) P (impossible event) =  P ( ?)  = 0 
(f) P(A – B) = P(A) – P(A ? B) = P(A ? B ) 
(g) ( ) ( ) ( ) ( ) P B A P B P A B P B A ? ? ? ? ? ? 
(h) ( ) ( ) 1 ( ) P A B P A B P A B ? ? ? ? ? ? 
(i) ( ) ( ) 1 ( ) P A B P A B P A B ? ? ? ? ? ? 
 
? Addition theorem for three events 
Let A, Band Cbe any three events associated with a random 
experiment, then 
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ).
P A B C P A P B P C P A B P B C P C A
P A B C
? ? ? ? ? ? ? ? ? ? ?
? ? ?
? Axiomatic Approach to Probability: 
Let S be a sample space containing elementary outcomes 
1 2
, ,...........
n
w w w
i.e. ? ?
1 2
, ,...........
n
S w w w ?
(i) 0 ( ) 1
i
P w ? ? for each 
i
w S ?
(ii) 
1 2
( ) ( ) ........... ( ) 1
n
P w P w P w ? ? ? ? 
(iii) ( ) ( )
i
P A P w ? ? for any event A containing elementary
events 
i
w . 
Verbal 
description of 
the event 
Equivalent Set 
Theoretic 
Notation 
Venn Diagram 
Not A 
' A or A 
A or B A B ?
 
A and B A B ?
A but not B 
(only A) 
A B ?
At least one of 
A, B or C 
A B C ? ?
Exactly one of A 
and B 
( ) ( ) A B A B ? ? ?
Neither A nor B 
( ) A B ?
Exactly two of 
A, B and C 
( )
( )
( )
A B C
A B C
A B C
? ? ?
? ?
? ? ?
All there of A, B 
and C 
A B C ? ?
The cards J, Q and K are called face cards. There are 12 face cards 
in a deck of 52 cards. 
There are 64 squares in a chess board i.e.32 white and 32 Black. 
Total n digit numbers = 
1
9 10 , 2
n
n
?
? ? 
e.g. there are 
2
9 10 900 ? ? . Three digit numbers.