Chapter 1 – Probability, counting methods and binomial
coefficients
1.1 Definitions and theorems
Def. 1.1 A random experiment is an experiment whose outcomes
cannot be predicted with certainty.
Examples: Tossing a coin. Tossing a pair of dice. Drawing a card from
a deck of 52 cards. The sequence of winning numbers in the Lotto draw.
Def. 1.2 A sample space of a random experiment is the collection of
all possible outcomes.
Examples
1 Tossing a die. Outcomes = S = {1,2,3,4,5,6}.
2 Drawing a winning number from Lotto. Outcomes = S = {0,1,2, ,49}
.
3 Drawing a card from a deck of 52 cards. Outcomes = { 2,3, . . . ,10,
jack, queen, king, ace for spades, club, diamonds and hearts } .
Def. 1.3 Each element of the sample space is called a sample point.
Def. 1.4 If the number of points are countable (finite) this samples
space is called a countable sample space.
Def 1.5 If the number of points are uncountable (infinite), then the
sample space is said to be a continuous sample space.
Examples
1 Countable sample space. Tossing a die where S = {1,2,3,4,5,6}.
Tossing a coin where S = {h, t} .
2 Continuous sample space. Measuring the temperature ( T ) in given
location, where S = {−5 T 32} .
1
,Def. 1.6 A subset A of the sample space S is said to be an event if it
belongs to a collection à of subsets of S satisfying the following three
rules:
(a) S à (b) If A à then Ac à (c) ) If Aj à then Aj à for
j 1.
The collection à is called an event space or a -field.
Theorem 1.1
à ( is an empty set which refers to an impossible event)
Proof
S Ã (a). Sc = Ã (b).
Theorem 1.2
If A1c Ã, A2c à then A1 A2 Ã
Proof
A1c à , A2c à (ii) and A1c A2c à (iii). Using (ii), it follows that
( A1c A2c ) c Ã. Using de Morgan’s law ( A1c A2c ) c = A1 A2 Ã.
Example
Describe the sample space S of rolling a pair of dice. Describe the event
A that the sum of numbers rolled is 7.
Solution
S = {(x, y) | x, y = 1, 2, 3, 4, 5, 6}
A = {(1, 6), (6, 1), (2, 5), (5, 2), (4, 3), (3, 4)}
Def. 1.7
2
,A function is a rule that associates each point in one set (A) of points
with one and only one point in another set of points (B). The sets A and
B are called the domain and counter domain respectively.
Let a A and b B . Then the function ( f ) can be written as b = f (a ) . The
set of all values of f (.) is called the range of f (.) .
Def. 1.8
Let S be the sample space of a random experiment. A probability
measure P : A* → [0,1] is a set function which assigns real numbers to
the various events of S satisfying
(P1) P ( A) 0 for all events A A* ,
(P2) P( S ) = 1 ,
(P3) P( Ak ) = Ak for A1 , A2 , mutually exclusive events of A* .
k =1 k =1
Theorem 1.3
P( ) = 0
Proof
Let A1 = S , Ai = for i = 2,3, . Then S = Ai and Ai A j = for i j.
i =1
1 = P( S ) = P( Ai ) = P( S ) + P( Ai ) = 1 + P( Ai ) = 1 + P( )
i =1 i =2 i =2 i =2
Therefore P( ) = 0 . Since any probability is
i =2
0, it follows that
P( ) = 0 .
Theorem 1.4
3
, Let A1 , A2 , , An be mutually exclusive events in à i.e. P ( Ai A j ) = for
n n
i j . Then P( Ai ) = P( Ai ) .
i =1 i =1
Proof
n
Let An+1 = An+2 = = . Then Ai = Ai à and
i =1 i =1
n n
P( Ai ) = P( Ai ) = P( Ai ) = P( Ai ) since P( ) = 0 .
i =1 i =1 i =1 i =1
For n = 2 the above result gives P( A1 A2 ) = P( A1 ) + P( A2 ) when A1 and A2
are mutually exclusive (disjoint).
Theorem 1.5
For any event A of the sample space S , P( Ac ) = 1 − P( A) .
Proof
A Ac = S and A Ac =
P( S ) = 1 = P( A Ac ) = P( A) + P( Ac ) P( Ac ) = 1 − P( A)
Theorem 1.6
If A and B Ã , then P( A B c ) = P( A) − P( A B) . The event A B c is also
written as A − B .
Proof
A = ( A B) ( A B c ) . Also ( A B) ( A B c ) = .
P( A) = P( A B) + P( A B c ) P( A B c ) = P( A) − P( A B) or
P( A − B ) = P( A) − P( A B )
The event A B can also be written as AB and therefore the result can
also be expressed as P( A − B) = P( A) − P( AB) .
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coefficients
1.1 Definitions and theorems
Def. 1.1 A random experiment is an experiment whose outcomes
cannot be predicted with certainty.
Examples: Tossing a coin. Tossing a pair of dice. Drawing a card from
a deck of 52 cards. The sequence of winning numbers in the Lotto draw.
Def. 1.2 A sample space of a random experiment is the collection of
all possible outcomes.
Examples
1 Tossing a die. Outcomes = S = {1,2,3,4,5,6}.
2 Drawing a winning number from Lotto. Outcomes = S = {0,1,2, ,49}
.
3 Drawing a card from a deck of 52 cards. Outcomes = { 2,3, . . . ,10,
jack, queen, king, ace for spades, club, diamonds and hearts } .
Def. 1.3 Each element of the sample space is called a sample point.
Def. 1.4 If the number of points are countable (finite) this samples
space is called a countable sample space.
Def 1.5 If the number of points are uncountable (infinite), then the
sample space is said to be a continuous sample space.
Examples
1 Countable sample space. Tossing a die where S = {1,2,3,4,5,6}.
Tossing a coin where S = {h, t} .
2 Continuous sample space. Measuring the temperature ( T ) in given
location, where S = {−5 T 32} .
1
,Def. 1.6 A subset A of the sample space S is said to be an event if it
belongs to a collection à of subsets of S satisfying the following three
rules:
(a) S à (b) If A à then Ac à (c) ) If Aj à then Aj à for
j 1.
The collection à is called an event space or a -field.
Theorem 1.1
à ( is an empty set which refers to an impossible event)
Proof
S Ã (a). Sc = Ã (b).
Theorem 1.2
If A1c Ã, A2c à then A1 A2 Ã
Proof
A1c à , A2c à (ii) and A1c A2c à (iii). Using (ii), it follows that
( A1c A2c ) c Ã. Using de Morgan’s law ( A1c A2c ) c = A1 A2 Ã.
Example
Describe the sample space S of rolling a pair of dice. Describe the event
A that the sum of numbers rolled is 7.
Solution
S = {(x, y) | x, y = 1, 2, 3, 4, 5, 6}
A = {(1, 6), (6, 1), (2, 5), (5, 2), (4, 3), (3, 4)}
Def. 1.7
2
,A function is a rule that associates each point in one set (A) of points
with one and only one point in another set of points (B). The sets A and
B are called the domain and counter domain respectively.
Let a A and b B . Then the function ( f ) can be written as b = f (a ) . The
set of all values of f (.) is called the range of f (.) .
Def. 1.8
Let S be the sample space of a random experiment. A probability
measure P : A* → [0,1] is a set function which assigns real numbers to
the various events of S satisfying
(P1) P ( A) 0 for all events A A* ,
(P2) P( S ) = 1 ,
(P3) P( Ak ) = Ak for A1 , A2 , mutually exclusive events of A* .
k =1 k =1
Theorem 1.3
P( ) = 0
Proof
Let A1 = S , Ai = for i = 2,3, . Then S = Ai and Ai A j = for i j.
i =1
1 = P( S ) = P( Ai ) = P( S ) + P( Ai ) = 1 + P( Ai ) = 1 + P( )
i =1 i =2 i =2 i =2
Therefore P( ) = 0 . Since any probability is
i =2
0, it follows that
P( ) = 0 .
Theorem 1.4
3
, Let A1 , A2 , , An be mutually exclusive events in à i.e. P ( Ai A j ) = for
n n
i j . Then P( Ai ) = P( Ai ) .
i =1 i =1
Proof
n
Let An+1 = An+2 = = . Then Ai = Ai à and
i =1 i =1
n n
P( Ai ) = P( Ai ) = P( Ai ) = P( Ai ) since P( ) = 0 .
i =1 i =1 i =1 i =1
For n = 2 the above result gives P( A1 A2 ) = P( A1 ) + P( A2 ) when A1 and A2
are mutually exclusive (disjoint).
Theorem 1.5
For any event A of the sample space S , P( Ac ) = 1 − P( A) .
Proof
A Ac = S and A Ac =
P( S ) = 1 = P( A Ac ) = P( A) + P( Ac ) P( Ac ) = 1 − P( A)
Theorem 1.6
If A and B Ã , then P( A B c ) = P( A) − P( A B) . The event A B c is also
written as A − B .
Proof
A = ( A B) ( A B c ) . Also ( A B) ( A B c ) = .
P( A) = P( A B) + P( A B c ) P( A B c ) = P( A) − P( A B) or
P( A − B ) = P( A) − P( A B )
The event A B can also be written as AB and therefore the result can
also be expressed as P( A − B) = P( A) − P( AB) .
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