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.
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