Page 1
Probability
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Probability
Equally likely events :
Outcomes of an event are said to be
‘equally likely’ when they have the same chance of occurring.
Ex: Rolling a die
Outcomes : 1, 2, 3, 4, 5 & 6. (All are equally likely to occur)
Impossible event : An event has no chance of occurrence ;
P ( Impossible event ) = 0
Ex: Getting the number 7 in a single roll of a die.
P( Getting 7 in a roll of a die ) = 0
Sure event : An event that has 100% chance of occurrence.
P ( sure event ) = 1
Ex: Getting number less than 7 in a single roll of a die.
P ( getting number less than 7 ) = 1
Mutually exclusive events ( Disjoint events ) :
Events A and B are said to be mutually exclusive if they do not have
any common point.
S = { 1, 2, 3, 4, 5, 6 }
A = { 4 } and B = { 1, 3, 5 }
A n B = ( empty set )
4
2
6
5
B
S
A
3
Exhaustive events : Two or more than two events
are said to be exhaustive events if their union is a sample space.
Ex: S = { 1, 2, 3, 4, 5, 6 }
A = { 3, 4, 5, 6 } and
B = { 1, 2, 3 }
A
n
B = { 1, 2, 3, 4, 5, 6 }
4
6
5
A
B
3
1
2
Complement events : Complementary events are those events
where probabilities of occurrence of one event exclude the
occurrence of the other.
Example :
Event (E) - Getting a head
Event (E) - Not getting a head (H) or Getting a tail (T)
P (E) = 1 - P (E) OR P (E) + P (E) = 1
Probability is the extent to which an event is likely to occur, measured by the ratio of the favourable cases to the whole number of cases possible.
It is classied into two categories:
Experimental or Empirical Probability
A probability is based on the outcome of an actual experiment &
adequate recording of the happening of an event.
A probability is based on the assumption about the outcome of an
event (E) rather than the outcome of an actual experiment.
In Probability theory, an event is a set of outcomes of an experiment to which a probability is assigned.
Theoretical or Classical Probability
< 7
Odd
Even
H T
PROBABILITY 35
Probability
Page 3
Probability
Equally likely events :
Outcomes of an event are said to be
‘equally likely’ when they have the same chance of occurring.
Ex: Rolling a die
Outcomes : 1, 2, 3, 4, 5 & 6. (All are equally likely to occur)
Impossible event : An event has no chance of occurrence ;
P ( Impossible event ) = 0
Ex: Getting the number 7 in a single roll of a die.
P( Getting 7 in a roll of a die ) = 0
Sure event : An event that has 100% chance of occurrence.
P ( sure event ) = 1
Ex: Getting number less than 7 in a single roll of a die.
P ( getting number less than 7 ) = 1
Mutually exclusive events ( Disjoint events ) :
Events A and B are said to be mutually exclusive if they do not have
any common point.
S = { 1, 2, 3, 4, 5, 6 }
A = { 4 } and B = { 1, 3, 5 }
A n B = ( empty set )
4
2
6
5
B
S
A
3
Exhaustive events : Two or more than two events
are said to be exhaustive events if their union is a sample space.
Ex: S = { 1, 2, 3, 4, 5, 6 }
A = { 3, 4, 5, 6 } and
B = { 1, 2, 3 }
A
n
B = { 1, 2, 3, 4, 5, 6 }
4
6
5
A
B
3
1
2
Complement events : Complementary events are those events
where probabilities of occurrence of one event exclude the
occurrence of the other.
Example :
Event (E) - Getting a head
Event (E) - Not getting a head (H) or Getting a tail (T)
P (E) = 1 - P (E) OR P (E) + P (E) = 1
Probability is the extent to which an event is likely to occur, measured by the ratio of the favourable cases to the whole number of cases possible.
It is classied into two categories:
Experimental or Empirical Probability
A probability is based on the outcome of an actual experiment &
adequate recording of the happening of an event.
A probability is based on the assumption about the outcome of an
event (E) rather than the outcome of an actual experiment.
In Probability theory, an event is a set of outcomes of an experiment to which a probability is assigned.
Theoretical or Classical Probability
< 7
Odd
Even
H T
PROBABILITY 35
Probability
}
{
n (S) = 12
Throwing a die and a coin
(H,1), (H,2), (H,3), (H,4), (H,5), (H,6)
(T,1), (T ,2), (T ,3), (T ,4), (T ,5), (T ,6)
n (S) = a
n
a - number of possible
outcomes of an event
n - number of events
n (A) - number of elements of an event A
n (S) - number of elements in Sample
Space
n (A)
n (S)
p (A) =
Sample Space
Tossing coins Rolling dice
Drawing cards
A coin
{
H, T
[ n (S) = 2
n
] [ n (S) = 6
n
] [ n (S) = 52 ]
2
1
6
1
2
2 6
2
2
3
}
n (S) = 2
T wo coins
{
HH, HT, TH, TT
}
n (S) = 4
Three coins
{ HHH, HHT, HTH, THH,
TTT, THT, HTT, TTH }
n (S) = 8
A die
{1, 2, 3, 4, 5, 6 }
n (S) = 6
Two dice
{ (1,1), (1,2), .......(1,6)
(2,1), ................(2,6)
............................
............................
............................
(6,1), ................(6,6)
}
n (S) = 36
Black / Red
n (A) = 26
Ace/ King/ Queen/
Jack/ Any number
n (A) = 4
Any Shape
n (A) = 13
A Face card
n (A) = 12
/ / /
Introduction to probability Ball and Card experiment and learn the
Basics of solving a probability problem.
Coin experiment and learn the basics of
solving a probability problem.
Problems based on Probability (Basics)
Scan the QR Codes to watch our free videos
Points to Remember
Probability of an event always lies between 0 and 1.
[ 0 = p ( x ) = 1 ]
p (happening an event) = 1 – p (Not happening that event)
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PLEASE KEEP IN MIND
p (sum of all possible outcomes) = 1
PROBABILITY 36
Happening of an Event provides with dierent Possible Outcomes. Here are the list of basic events and their outcomes which will help you to
start and understand the topic probability in a better manner.
Read More