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PROBABILITY

295

PROBABILITY

15

The theory of probabilities and the theory of errors now constitute
a formidable body of great mathematical interest and of great
practical importance
...
S
...
1 Introduction
In Class IX, you have studied about experimental (or empirical) probabilities of events
which were based on the results of actual experiments
...
e
...
455 and
1000
that of getting a tail is 0
...
(Also see Example 1, Chapter 15 of Class IX Mathematics
Textbook
...
For this reason, they are called experimental or empirical
probabilities
...
Moreover,
these probabilities are only ‘estimates’
...


Based on this experiment, the empirical probability of a head is

In Class IX, you tossed a coin many times and noted the number of times it turned up
heads (or tails) (refer to Activities 1 and 2 of Chapter 15)
...

For example, the eighteenth century French naturalist Comte de Buffon tossed a
coin 4040 times and got 2048 heads
...
e
...
507
...
E
...
The experimental probability of getting a head, in this case,

5067
= 0
...
Statistician Karl Pearson spent some more time, making 24000
10000
tosses of a coin
...
5005
...
5 , i
...
,

1
, which is what we call the theoretical probability of getting a head (or getting a
2
tail), as you will see in the next section
...


15
...

When we speak of a coin, we assume it to be ‘fair’, that is, it is symmetrical so
that there is no reason for it to come down more often on one side than the other
...
By the phrase ‘random toss’,
we mean that the coin is allowed to fall freely without any bias or interference
...
We can reasonably assume that each
outcome, head or tail, is as likely to occur as the other
...


PROBABILITY

297

For another example of equally likely outcomes, suppose we throw a die
once
...
What are the possible outcomes?
They are 1, 2, 3, 4, 5, 6
...
So
the equally likely outcomes of throwing a die are 1, 2, 3, 4, 5 and 6
...

Suppose that a bag contains 4 red balls and 1 blue ball, and you draw a ball
without looking into the bag
...
So, the outcomes
(a red ball or a blue ball) are not equally likely
...
So, all experiments do not necessarily
have equally likely outcomes
...

In Class IX, we defined the experimental or empirical probability P(E) of an
event E as

Number of trials in which the event happened
Total number of trials
The empirical interpretation of probability can be applied to every event associated
with an experiment which can be repeated a large number of times
...
Of course, it worked well in coin tossing or die throwing
experiments
...
The assumption of equally likely outcomes (which is
valid in many experiments, as in the two examples above, of a coin and of a die) is one
such assumption that leads us to the following definition of probability of an event
...

We will briefly refer to theoretical probability as probability
...

Probability theory had its origin in the 16th century when
an Italian physician and mathematician J
...

Since its inception, the study of probability has attracted
the attention of great mathematicians
...
de Moivre (1667 – 1754), and
Pierre Simon Laplace are among those who made significant
contributions to this field
...
Pierre Simon Laplace
(1749 – 1827)
In recent years, probability has been used extensively in
many areas such as biology, economics, genetics, physics,
sociology etc
...

Example 1 : Find the probability of getting a head when a coin is tossed once
...

Solution : In the experiment of tossing a coin once, the number of possible outcomes
is two — Head (H) and Tail (T)
...
The number of
outcomes favourable to E, (i
...
, of getting a head) is 1
...
Kritika takes out a ball from the bag without looking into it
...
So, it is equally
likely that she takes out any one of them
...

Now, the number of possible outcomes = 3
...

So,
Similarly,

1
3
1
1
(ii) P(R) =
and (iii) P(B) = ⋅
3
3
P(Y) =

Remarks :
1
...
In Example 1, both the events E and F are elementary events
...

2
...
This is true in general also
...
(i) What is the probability of getting a
number greater than 4 ? (ii) What is the probability of getting a number less than or
equal to 4 ?
Solution : (i) Here, let E be the event ‘getting a number greater than 4’
...
Therefore, the number of outcomes favourable to E is 2
...

P(E) = P(number greater than 4) =

Number of possible outcomes = 6
Outcomes favourable to the event F are 1, 2, 3, 4
...

Therefore,

P(F) =

4
2
=
6
3

300

MATHEMATICS

Are the events E and F in the example above elementary events? No, they are
not because the event E has 2 outcomes and the event F has 4 outcomes
...

P(E) + P(F) =

(1)

From (i) and (ii) of Example 3, we also get
P(E) + P(F) =

1 2
+ =1
3 3

(2)

where E is the event ‘getting a number >4’ and F is the event ‘getting a number ≤ 4’
...

In (1) and (2) above, is F not the same as ‘not E’? Yes, it is
...

So,
P(E) + P(not E) = 1
i
...
,

P(E) + P( E ) = 1,

which gives us P( E ) = 1 – P(E)
...

We also say that E and E are complementary events
...
These
outcomes are 1, 2, 3, 4, 5 and 6
...
e
...
In other words,
getting 8 in a single throw of a die, is impossible
...
Such an
event is called an impossible event
...
So, the number of
favourable outcomes is the same as the number of all possible outcomes, which is 6
...
Such an event
is called a sure event or a certain event
...
Therefore,
0 ≤ P(E) ≤ 1
Now, let us take an example related to playing cards
...
Clubs and spades are of black
colour, while hearts and diamonds are of red colour
...
Kings, queens and jacks are called face
cards
...
Calculate the
probability that the card will
(i) be an ace,
(ii) not be an ace
...

(i) There are 4 aces in a deck
...

The number of outcomes favourable to E = 4
The number of possible outcomes = 52

(Why ?)

4
1
=
52 13
(ii) Let F be the event ‘card drawn is not an ace’
...
Therefore, we can also calculate P(F) as
follows: P(F) = P( E ) = 1 – P(E) = 1 −

1 12
= ⋅
13 13

Example 5 : Two players, Sangeeta and Reshma, play a tennis match
...
62
...

The probability of Sangeeta’s winning = P(S) = 0
...
62 = 0
...
What is the probability that both will
have (i) different birthdays? (ii) the same birthday? (ignoring a leap year)
...
Now, Hamida’s birthday can also be any day of 365 days in the year
...

(i) If Hamida’s birthday is different from Savita’s, the number of favourable outcomes
for her birthday is 365 – 1 = 364
So, P (Hamida’s birthday is different from Savita’s birthday) =

364
365

(ii) P(Savita and Hamida have the same birthday)
= 1 – P (both have different birthdays)
= 1−
=

364
365

1
365

[Using P( E ) = 1 – P(E)]

PROBABILITY

303

Example 7 : There are 40 students in Class X of a school of whom 25 are girls and 15
are boys
...
She
writes the name of each student on a separate card, the cards being identical
...
She then draws one card from the
bag
...

(i) The number of all possible outcomes is 40
The number of outcomes favourable for a card with the name of a girl = 25 (Why?)

25 5
=
40 8
(ii) The number of outcomes favourable for a card with the name of a boy = 15 (Why?)
Therefore, P (card with name of a girl) = P(Girl) =

Therefore, P(card with name of a boy) = P(Boy) =

15 3
=
40 8

Note : We can also determine P(Boy), by taking
P(Boy) = 1 – P(not Boy) = 1 – P(Girl) = 1 −

5 3
=
8 8

Example 8 : A box contains 3 blue, 2 white, and 4 red marbles
...
Therefore, the
number of possible outcomes = 3 +2 + 4 = 9

(Why?)

Let W denote the event ‘the marble is white’, B denote the event ‘the marble is blue’
and R denote the event ‘marble is red’
...


and

(iii) P(R) =

4
9

304

MATHEMATICS

Example 9 : Harpreet tosses two different coins simultaneously (say, one is of Re 1
and other of Rs 2)
...
When two coins are tossed
simultaneously, the possible outcomes are (H, H), (H, T), (T, H), (T, T), which are all
equally likely
...
Similarly (H, T) means head up on the first coin and tail up
on the second coin and so on
...
(Why?)
So, the number of outcomes favourable to E is 3
...
e
...

There are many experiments in which the outcome is any number between two
given numbers, or in which the outcome is every point within a circle or rectangle, etc
...
So, the definition of (theoretical) probability
which you have learnt so far cannot be applied in the present form
...

What is the probability that the music will stop within the first half-minute after starting?
Solution : Here the possible outcomes are all the numbers between 0 and 2
...
15
...


Fig
...
1
* Not from the examination point of view
...

The outcomes favourable to E are points on the number line from 0 to

1

...

2
2
Since all the outcomes are equally likely, we can argue that, of the total distance

The distance from 0 to 2 is 2, while the distance from 0 to

of 2, the distance favourable to the event E is

So,

1

...
15
...
What is the probability that it crashed inside the
lake shown in the figure?

Fig
...
2
Solution : The helicopter is equally likely to crash anywhere in the region
...
5 × 9) km2 = 40
...


306

MATHEMATICS

Area of the lake = (2
...
5 km2
Therefore, P (helicopter crashed in the lake) =

7
...
5 405 27

Example 12 : A carton consists of 100 shirts of which 88 are good, 8 have minor
defects and 4 have major defects
...
One shirt is drawn at random from the carton
...
Therefore,
there are 100 equally likely outcomes
...
e
...
88
100

(ii) The number of outcomes favourable to Sujatha = 88 + 8 = 96
So, P (shirt is acceptable to Sujatha) =

(Why?)

96
= 0
...
Write
down all the possible outcomes
...
The same is true when the blue die shows ‘2’, ‘3’, ‘4’, ‘5’ or
‘6’
...


PROBABILITY

307

4
6

5

1

5

6

(1, 1)

(1, 2)

(1, 3)

(1, 4)

(1, 5)

(1, 6)

2

(2, 1)

(2, 2)

(2, 3)

(2, 4)

(2, 5)

(2, 6)

3

(3, 1)

(3, 2)

(3, 3)

(3, 4)

(3, 5)

(3, 6)

(4, 1)

(4, 2)

(4, 3)

(4, 4)

(4, 5)

(4, 6)

5

(5, 1)

(5, 2)

(5, 3)

(5, 4)

(5, 5)

(5, 6)

6

5

4

4

6

3

1
4

2

(6, 1)

(6, 2)

(6, 3)

(6, 4)

(6, 5)

(6, 6)

Fig
...
3
Note that the pair (1, 4) is different from (4, 1)
...

(i) The outcomes favourable to the event ‘the sum of the two numbers is 8’ denoted
by E, are: (2, 6), (3, 5), (4, 4), (5, 3), (6, 2) (see Fig
...
3)
i
...
, the number of outcomes favourable to E = 5
...
15
...

So,

P(F) =

0
=0
36

(iii) As you can see from Fig
...
3, all the outcomes are favourable to the event G,
‘sum of two numbers ≤ 12’
...
1
1
...



...
Such an event is

(iii) The probability of an event that is certain to happen is
is called

...
Such an event

(iv) The sum of the probabilities of all the elementary events of an experiment is

...


and less than or

2
...

(i) A driver attempts to start a car
...

(ii) A player attempts to shoot a basketball
...

(iii) A trial is made to answer a true-false question
...

(iv) A baby is born
...

3
...
Which of the following cannot be the probability of an event?
(A)

2
3

(B) –1
...
7

5
...
05, what is the probability of ‘not E’?
6
...
Malini takes out one candy without
looking into the bag
...
It is given that in a group of 3 students, the probability of 2 students not having the
same birthday is 0
...
What is the probability that the 2 students have the same
birthday?
8
...
A ball is drawn at random from the bag
...
A box contains 5 red marbles, 8 white marbles and 4 green marbles
...
What is the probability that the marble taken out will be
(i) red ? (ii) white ? (iii) not green?

PROBABILITY

309

10
...
If it is equally likely that one of the coins will fall out when the bank is turned
upside down, what is the probability that the coin (i) will be a 50 p coin ? (ii) will not be
a Rs 5 coin?
11
...
The
shopkeeper takes out one fish at random from a
tank containing 5 male fish and 8 female fish (see
Fig
...
4)
...
A game of chance consists of spinning an arrow
which comes to rest pointing at one of the numbers
1, 2, 3, 4, 5, 6, 7, 8 (see Fig
...
5 ), and these are equally
likely outcomes
...
A die is thrown once
...
15
...
15
...


14
...
Find the probability of getting
(i) a king of red colour
(ii) a face card
(iii) a red face card
(iv) the jack of hearts
(v) a spade
(vi) the queen of diamonds
15
...
One card is then picked up at random
...
12 defective pens are accidentally mixed with 132 good ones
...
One pen is taken out at random from
this lot
...

17
...
One bulb is drawn at random from the lot
...
Now one bulb
is drawn at random from the rest
...
A box contains 90 discs which are numbered from 1 to 90
...


310

MATHEMATICS

19
...
What is the probability of getting (i) A?

(ii) D?

20*
...
15
...
What is
the probability that it will land inside the circle with diameter 1m?
3m

2m

Fig
...
6
21
...
Nuri
will buy a pen if it is good, but will not buy if it is defective
...
What is the probability that
(i) She will buy it ?
(ii) She will not buy it ?
22
...
(i) Complete the following table:
Event :
‘Sum on 2 dice’
Probability

2

3

4

1
36

5

6

7

8

9

10

5
36

11

12
1
36

(ii) A student argues that ‘there are 11 possible outcomes 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and
1

...
Therefore, each of them has a probability
11
Justify your answer
...
A game consists of tossing a one rupee coin 3 times and noting its outcome each time
...
e
...
Calculate the probability that Hanif will lose the game
...
A die is thrown twice
...


PROBABILITY

311

25
...

(i) If two coins are tossed simultaneously there are three possible outcomes —two
heads, two tails or one of each
...
Therefore, the probability of getting an odd number is
...
2 (Optional)*
1
...
Each is equally likely to visit the shop on any day as on another day
...
A die is numbered in such a way that its faces show the numbers 1, 2, 2, 3, 3, 6
...
Complete the following
table which gives a few values of the total score on the two throws:

Number in second throw

+

1

Number in first throw
2
2
3

1

2

3

3

4

4

7

2

3

4

4

5

5

8

2

3

6

5

3
3
6

5
7

8

8

9
9

9

12

What is the probability that the total score is
(i) even?

(ii) 6?

(iii) at least 6?

3
...
If the probability of drawing a blue ball
is double that of a red ball, determine the number of blue balls in the bag
...
A box contains 12 balls out of which x are black
...
Find x
...


312

MATHEMATICS

5
...
If a marble is drawn at
random from the jar, the probability that it is green is

2
⋅ Find the number of blue balls
3

in the jar
...
3 Summary
In this chapter, you have studied the following points :
1
...

2
...

3
...

4
...

5
...
An event having only one outcome is called an elementary event
...

7
...
E and E are called
complementary events
...
As the number of trials in an experiment, go on
increasing we may expect the experimental and theoretical
probabilities to be nearly the same

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