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# Probability - Chapter Notes, Class 9 Mathematics Class 9 Notes | EduRev

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## Class 9 : Probability - Chapter Notes, Class 9 Mathematics Class 9 Notes | EduRev

``` Page 1

PROBABILITY
The word Probability is commonly used to represent
uncertainty of an event.

Example
(i) Probably, it may rain today.
(ii) Probably, he may stand first in the class.

So, Probability is used to mention the chances or possibilities
of a thing to occur. In other words, it represents the
chances/possibilities for an event to occur out of an
operation/experiment.

Example
If we throw a dice, then we may obtain any number out of the
six numbers of dice. In this case, throwing the dice is the
experiment/operation and getting a particular number is an
event.
To study the ‘Probability’ mathematically, we will first study
some terms related to ‘Probability’.

1. Experiment
The word ‘experiment’ means, the operation which produces
the result.

The experiments are divided into two categories.

(i) Deterministic Experiment
Deterministic experiments are those experiments, which
on repeating under same conditions produce same result.

Example
Science or engineering events under same conditions
produce same or almost same results.

(ii)  Probabilistic / Random Experiments
Probabilistic or random experiments are those
experiments, which on repeating under same conditions
do not produce same results. In other words, these
experiments produce one single result out of several
possible results.

Example
Throwing a dice under same conditions will produce
different results each time or maximum times.

2. Event
The result of an experiment is called event. These are
following types of events: -

(i) Elementary Event
The basic result of a random experiment is called
elementary event. That means, the final single result
obtain by an experiment is called elementary events.

Example
(i) On tossing a can randomly, we either get head or tail.
Then,
T = Getting T (tail)

So, H and T are elementary events, which are the results
of random experiment tossing the coin.

(ii) On tossing two coins simultaneous, we get different
Then,
HT = Getting head (H) on first and tail (T) on second.
HH = Getting head (H) on first and head (H) on second
TH = Getting tail (T) and first and head (H) on second
TT = Getting tail (T) on first and tail (T) on second

So, HT, HH, TH and TT are elementary events.

(iii) On throwing a dice, we get different numbers
Then,
-
1
N Getting Number 1
-
2
N Getting number 2
-
3
N Getting number 3
-
4
N Getting number 4
-
5
N Getting number 5
-
6
N Getting number 6

So
5 4 3 2 1
N , N , N , N , N and
6
N are elementary events.

(ii) Compound Event
The result of a random experiment made up of more than one
elementary event is called compound event.

Example
On throwing a dice, getting a number is elementary event. In
such case, getting an event number is a compound event,
which is made of three elementary events, i.e. getting number
1, getting number 3 and getting number 5.

(iii) Possible Event
All the possible results, which may produced by the experiment
are called possible events

Example
(i) On tossing a coin, we may get 2 possible results.
T – Getting tail (T)

So, number of possible events is 2, i.e. H and T.

(ii) On throwing a dice, we may get 6 possible results.
-
1
N Getting number 1
-
2
N Getting number 2
-
3
N Getting number 3
-
4
N Getting number 4
-
5
N Getting number 5
-
6
N Getting number 6.
So, number of possible events is 6, i.e.
1
N ,
2
N ,
3
N ,
4
N ,
5
N ,
6
N

Page 2

PROBABILITY
The word Probability is commonly used to represent
uncertainty of an event.

Example
(i) Probably, it may rain today.
(ii) Probably, he may stand first in the class.

So, Probability is used to mention the chances or possibilities
of a thing to occur. In other words, it represents the
chances/possibilities for an event to occur out of an
operation/experiment.

Example
If we throw a dice, then we may obtain any number out of the
six numbers of dice. In this case, throwing the dice is the
experiment/operation and getting a particular number is an
event.
To study the ‘Probability’ mathematically, we will first study
some terms related to ‘Probability’.

1. Experiment
The word ‘experiment’ means, the operation which produces
the result.

The experiments are divided into two categories.

(i) Deterministic Experiment
Deterministic experiments are those experiments, which
on repeating under same conditions produce same result.

Example
Science or engineering events under same conditions
produce same or almost same results.

(ii)  Probabilistic / Random Experiments
Probabilistic or random experiments are those
experiments, which on repeating under same conditions
do not produce same results. In other words, these
experiments produce one single result out of several
possible results.

Example
Throwing a dice under same conditions will produce
different results each time or maximum times.

2. Event
The result of an experiment is called event. These are
following types of events: -

(i) Elementary Event
The basic result of a random experiment is called
elementary event. That means, the final single result
obtain by an experiment is called elementary events.

Example
(i) On tossing a can randomly, we either get head or tail.
Then,
T = Getting T (tail)

So, H and T are elementary events, which are the results
of random experiment tossing the coin.

(ii) On tossing two coins simultaneous, we get different
Then,
HT = Getting head (H) on first and tail (T) on second.
HH = Getting head (H) on first and head (H) on second
TH = Getting tail (T) and first and head (H) on second
TT = Getting tail (T) on first and tail (T) on second

So, HT, HH, TH and TT are elementary events.

(iii) On throwing a dice, we get different numbers
Then,
-
1
N Getting Number 1
-
2
N Getting number 2
-
3
N Getting number 3
-
4
N Getting number 4
-
5
N Getting number 5
-
6
N Getting number 6

So
5 4 3 2 1
N , N , N , N , N and
6
N are elementary events.

(ii) Compound Event
The result of a random experiment made up of more than one
elementary event is called compound event.

Example
On throwing a dice, getting a number is elementary event. In
such case, getting an event number is a compound event,
which is made of three elementary events, i.e. getting number
1, getting number 3 and getting number 5.

(iii) Possible Event
All the possible results, which may produced by the experiment
are called possible events

Example
(i) On tossing a coin, we may get 2 possible results.
T – Getting tail (T)

So, number of possible events is 2, i.e. H and T.

(ii) On throwing a dice, we may get 6 possible results.
-
1
N Getting number 1
-
2
N Getting number 2
-
3
N Getting number 3
-
4
N Getting number 4
-
5
N Getting number 5
-
6
N Getting number 6.
So, number of possible events is 6, i.e.
1
N ,
2
N ,
3
N ,
4
N ,
5
N ,
6
N

(iv) Favourable Events
All the results, which may satisfy the given condition are called
favourable events.

Example
(i) On tossing a coin and getting the tail, we have one
favourable event.
T = Getting tail

(ii) On throwing a dice and getting a number greater than or
equal to 5, we have two favourable events.
-
5
N Getting number 5
-
6
N Getting number 6

3. Negation Of Event
If E is an event, then negation of event E is NotE. In other
words, the unfavourable results are called negation of event E.
Negation of event is represented by a bar on the event. So, if
event is E, then negation of event is E .

Example
On throwing a dice to get number 4.
Event – Getting number 4, i.e.
4
N
So,
Negation Of Event – Getting a number other than 4, i.e.
2 1
N , N ,
5 3
N , N and
6
N

Probability
Now, we will study Probability mathematically. That means,
we will represent Probability in terms of numeric values.

Let all the possible events = n
Let all the favourable events = m
Probability = P(A)
Then,
n
m
) A ( P =

So, if there are n possible events of a random experiment, out
of which there are m favourable events, then probability is the
ratio of possible events and favourable events.

1. If all possible events satisfy the condition, then the
event is sure to happen.
So, possible events = n
Favourable events = m
Then, probability
P(A) =
n
m

?
n
n
) A ( P =       [Possible Events = Favourable Event]
? 1 ) A ( P =

? Probability of an event sure to happen = 1

2. If no possible event satisfy the condition, then the
event is sure not to happen.
So, possible events = n
Favourable event = m = 0
Then, probability
n
m
) A ( P =
?
n
0
) A ( P =          [m = 0]
? 0 ) A ( P =

? Probability of an event sure not to happen =0

3. If there are n possible events and m favourable
events, then there are  m n - unfavourable events.
So, possible events = n
Favourable events = m
Unfavourable events = n – m
Then, probability
n
m
) A ( P =        ………………… (i)
Also, probability of negation of event
m
m n
) A ( P
-
=        ………………… (ii)
So,
n
m n
n
m
) A ( P ) A ( P
-
+ = +
?
n
m n m
) A ( P ) A ( P
- +
= +
?
n
n
) A ( P ) A ( P = +
? 1 ) A ( P ) A ( P = +
That means,
) A ( P 1 ) A ( P - =     Or
) A ( P 1 ) A ( P - =

Example
A card is drawn from a well-shuffled deck of playing cards.
Find the probability of drawing: -
(i) A face card.
(ii) A red face card.

We have,
Total number of possible events ) n ( = Total Cards = 52

(i) Ace, king, queen and jack are called face cards.
There are 4 aces, 4 kings, 4 queens, 4 jacks in a pack of cards.
Then,
Number of fvourable events (m) = Total Face Cards
= 4 + 4 + 4
= 12
So, Probability of getting a face card
n
m
) A ( P =
=
12
52

=
3
13

(ii) There are 2 red aces, 2 red kings, 2 red queens and 2 red
jacks in a pack of cards.
Then,
Number of favourable events (m) =Total Number Of Red Faces
= 2 + 2 + 2 + 2
=8
So, probability of getting a red face card
n
m
) A ( P =
=
52
8

=
13
2

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