UNIT-VIll: PROBABILITY

21) PROBABILITY

experience,
people often make
statements like:
nscod on past
This boy is likely to top the school in the ICSE examination this year.
Mostprobably it will rain today.
There is a good chance of India winning the cricket match against Bangladesh.
All these words -likely, probably, chance involve an element of uncertainty. Probability measures
nuimerically the possibility of an event happening. A numerical value is given to an event to show how

likely the event

can happen.
Some Basic Terms
AnExperiment is an activity which produces well defined outcomes (or results).
A Random experiment is an experiment in which all possible outcomes of the experiment are
known but which outcome will occur cannot be predicted. The results can be different even in
identical situations.

Tossing of a coin, rolling a dice or drawing a card from a pack of 52 playing cards are all random
experiments.
Sample space: The set of all possible outcomes of a random experiment is known as the sample
space.
An Event is a subset of the sample space. It is a well-defined collection of some specific outcomes
of an experiment. For example, in the experiment of tossing 3 coins simultaneously, the event of
getting exactly 2 heads is {HHT, HTH, THH}.
An Elementary event is an event with only one favourable outcome. For example,
when 5 identical cards are marked with numbers 1 to 5 picking a card with number 3 is an
elementary event.
A Compound event is an event which has more than one favourable outcome. For example,
in the above example picking an odd numbered card has favourable outcomes {1, 3, 5}.
then E is the event non-occurring.
Complementary events: If E is an event happening,
laking the above example of 5 cards, picking an even numbered card E = (2, 4 , E = {1, 3, 5), t h e n

EUE S = Sample space

P(E) + P(E) = 1
said to be mutually exclusive if they both
aly exclusive events: Two events A and B are cannot
Occur together or the intersection of the two events is empty.

ANB=o and P(A UB) = P(A) + P(6)

For and tail cannot show up together.
dnple, when a coin is tOssed both head

Occurrence of an event
An ev c t occurs when the outcome of an experiment satisties the condition mentioned in the event.
For example, wh tossed if an event E is named as getting two tails and if the pair of
when two coins are
coins shows
w S two tails, we say that event E has occurred.
 Favourable outcomes
The outcomes which ensure the occurrence of an event are called favourable outcomes. For ex
the favourable outcomes for an event E of getting a sum of 5 when two dice are rolled are ja example,
(4, 1), (2, 3), (3, 2)}. 11,4
Here (1, 4) means showing up 1 on the first dice and 4 on the second dice.

Equally likely outcomes

Two or more outcomes are called equally likely outcomes if any one of the outcomes can occur with
any preference to the other.
Examples:
)Whena coin is tossed, it is equally likely that the coin shows up either head or tail.
(i) When a dice is rolled, each of the numbers 1, 2, 3, 4, 5, 6 is equally likely to ocur.
in) When card
a is selected from pack of well shuffled playing cards,
a an event of getting a clubis
equally likely as getting a heart or a spade or a diamond.
But bag when there
in a

getting a green ball.

are 4 red and 6 green balls, the outcome of getting a red ball is not same as
Sure Event
An event which always happens is a sure event. The probability of a sure event is 1.
Impossible Event
An event which never happens is an impossible event. The probability of an impossible event is zero
For example, when a dice is rolled, probability of getting a nunmber more than 6 is an impossible
event
Classical/Theoretical Probability
Here predict what will happen without
we

outcomes are equally likely to happen.

actually performing the experiment. We assume that all
If E is an event, then Probability of the event E happening
outcomes toE
PE) =No. of favourable
Total no. of possible outcomes
The probability has to satisfy 3 axioms:
(i) 0 <PE) < 1
The probability of event is a number between 0 and 1 both inclusive.

(i) P(S) =1
The probability of the sample space S considered as an event NOTE
has to be 1. It is called a sure event. For example, It probability of an event happening
1s P,then the probability of the event
when a dice is rolled, the probability of getting a
positive not happening is 1 -P.
number less than 7 is 1.
(i) For any sequence of mutually exclusive events E, Ez, Es their sum of probability is 1. r

example, when a dice is rolled, then

P(1) + P2) + P(3) + P(4) + P(5) + P(6) =1

Empirical/Experimental Probability
Here probability of an event E is based on what actually happens in the experiment.
D No. of trials in which the event occurs
Total no. of trials

ICSE Mathematics 10
(350
 c o i n s are
22 coins
led 100 times and the results
rolle
whern are as follows:
e.g,
0 1
N o . o fh e a d s
2 Total
No. o f t r i a l s
14 56 30 100
14
robability of getting one head = 6
Here, pr 100 25 Fig. 21.1

NOTE
2. A pack of playing cards has 52 cards.
Total no. of
Random Experiment 52 Cards
outcomes

(i) When a coin is tossed 2 2 26 Black Cards

26 Red Cards

(it) When 2 coins are tossed 22 4

(iit) When 3 coins are tossed 23 8

13 Diamonds 13 Hearts 13 Clubs 13 Spades

(io) When a dice is rolled 6 =6 Fig. 21.2

These are 4 suits each consisting of 2 to 10 numbered
() When 2 dice are rolled 62 36
cards and Jack, Queen, King-face cards.
I n a pack there are 12 face cards.
(vi) When 3 dice are rolled 63 216 Each suit has an Ace, which is neither a face card nor
a number card.

Example 1: If 3 unbiased coins are tossed, what is the probability of getting

(i) one head (ii) 2 heads

(i1) note at least 2 m e a n s or more favourable outcomes (io) at most one head?

Solution:
Totalnumber of outcomes 23 8 which are {(HHH), (HHT), (HTH), (THH), (TITH), (THT), (HTT), (TTT)}.
() Favourable outcomes for 1 head {(TTH), (THT), (HTT) =

No. of favourable outcomes

Probability of getting 1 head= Total no. of outcomes

(1) Favourable outcomes for 2 heads =

{(HHT), (HTH), (THH)}
3
Probability of getting 2 heads
=

un) Favourable outcomes for at least 2 heads = {(HHT), (HTH), (THH), (HHH

2 heads 3 heads
of getting at least 2 heads
= or =

Probability 8 2
Favourable outcomes for at most 1 head
=
{(TTT), (HTT), (THT), (TTH)
Probability of getting at most 1 head = 0 heads or 1 head =8 2

Example 2: If two dice are thrown simultaneously, what is the probability of getting on the uppermost
faces of the dice?
) a doublet (same number on both) (11) a sum of 7 (ii) a product of 12
(v) 4 or5 (vi) More than 10.
Gio) both are odd numbers.

Probability 351
 Solution:
Total no. of outcomes = 62 = 36

1) Favourable outcomes for doublet =

{(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}
P(doublet) =36 6
(11) Favourable outcomes for a sum of 7 ={(1, 6), (6, 1), (2, 5), (5, 2), (3, 4), (4, 3)
P(sum7) 6
(ii) Favourable outcomes for a product of 12 = {(2, 6), (6, 2), (3, 4), (4 3)

P(product 12) ==
36
(iv) Favourable outcomes for both are odd numbers = {(1, 1), (1, 3), (1, 5), (3, 1), (3, 3), (3, 5), (5. 11

(5, 3), (5, 5)}

P(both odd) = -
B6
() The sum of 2 numbers is 4 = (1 + 3), (2 + 2), (3 + 1);

5 (1 +4), (2+3), (3 +2), (4 + 1) =

7 favourable outcomes
Total no. of outcomes = 6 = 36

Probability = 36
(oi) More than 10 = 11 or 12

Sum 11 {(6 + 5), (5 +6)) =2 outcomes; Sum 12 = (6 +6) = 1 outcome

Probability =
Example 3: Some identical cards are numbered from 2 to 25 and well shuffled. When a card is drawn
randomly, what is the probability that the card has
() a prime number (i) a multiple of 2 and 3 (iii) not a multiple of 7.
Solution: From 1 to 25, there are 25 cards but 1 is not there.
Total no. of cards =24. or no. of cards = (25-2) +1
() A prime number = {2, 3,5, 7, 11, 13, 17, 19, 23)

P(prime)=
(i) A multiple of 2 and 3 =
a
multiple of 6 {6, 12, 18, 24)
P(multiple of 2and 3) =

(in) A multiple of 7 = {7, 14, 21), i.e., 3 cards but not a multiple of 7 = 24 - 3 21 cards.

. P(not a multiple of 7) =

Example 4: If all face cards removed from

the
are a pack of 52 playing cards and well shuffled, then what
l
probability picking a
of card with
() a red even number (ii) a club odd number (ii) a Jack (10) a prime number.
Solution: 12 face cards are removed from the pack.
Total number of cards 52 -

12 =40.
352 ICSE Mathematics 10
 bers
E v e nn u m b e r s {2, 4, 6, 8, 10} x 2 red suits 10
P(red e v e n ) = 1
=
favourable cards
40 4
A club odd number = {3, 5, 7, 9
i) i.e., 4 favourable cards
4
P(club odd) =
40 10
are no
Alack, there Jacks as all face cards are removed.

P(Jack)=00
40
Gir) A prime number = {2, 3 , 5 , 7} x 4 suits
i.e., 16 favourable cards
16 2
P(prime) =
40
vample 5: If All even numbered cards are removed from a pack of 52 playing cards. What 1s the
Exa
probability that a card picked up is
(i) a face card
(ii) a multiple of 3 of clubs
(ii) a red multiple of 5 io) a prime numbered card.
Solution: Even number cards ={2, 4, 6, 8, 10) x 4 suits i.e., 20 cards are removed.
N o . of cards left = 52 20 = 32.

() No. of face cards =12

P(face card) =

(i) A multiple of 3 of clubs = {3, 9}

Note that 6 is removed.

1
P(multiple of 3 of clubs) =

16
(1) A red multiple of 5 - {5 of Hearts, 5 of diamonds) = 2 cards

Note that even number 10 is removed.

1
of 5)
P(red multiple of 5) o =

32 16
(10) A prime numbered card = {3, 5, 7}x 4 suits = 12 cards

Note that prime number 2 is removed.

12
P(prime) 32
xample 6: A bag contains 5 red balls and some green balls. If the probability of drawing a green
Dall is double
that of a red ball, find the number of green balls in the bag.

green balls and 5 red

otion: Let there be x balls
Total no. of balls =x + 5
No. of green balls
P(green ball) P(red ball) =
No. of red balls 5
Total no. of balls X+5 Total no. of balls +5

P(green ball) = 2 P(red ball)

2
x = 2 x 5 10
No. of
green balls =
10.

Probability 353
 made with 3, 5, 7 and 9,
what is the probability that the
Example 7: If two-digit numbers are
r
mber is
()greaterthan 55 prime number.
(ii) a

59), (73, 75, 77, 79), (93, 95, 97, 9911

Solution: Two digit numbers =
{(33, 35, 37, 39), (53, 55, 57,
Total no. = 16
97, 99
() Numbers greater than 55 (57, 59, 73, 75, 77, 79, 93, 95,
P(number > 55) = 1
16 8
(17) Prime Numbers = {37, 53, 59, 73, 79, 97}
3
P(a prime number)= 16 8

Example office works for 5 days in a week

8: If an (Monday to Friday) and it two employees of the
that it is
ofice remain absent in the same week, what is the probability
ii) different days (iii) consecutive days?
(i) the same day
Solution:
Total no. of working days for both in a week = 5 = 25

(1) Same day =

{(Mon, Mon), (Tue, Tue), (Wed, Wed), (Thurs, Thurs) (Fri, Fri)}
P(sameday) =

(it) P(different days) =1 - P(absent on the sanme day) = 1

(ii) Consecutive days = {(Mon, Tue), (Tue, Wed), (Wed, Thurs), (Thurs, Fri), (Tue, Mon), (Wed, Tue)
(Thurs, Wed), (Fri, Thurs)}
8
P(consecutive days) 25
the probability that in January may have 5 Mondays if it is a
Example 9: Find a
year
(i) leap year (ii) non-leap year.
Solution:
(1) If 1st Jan is a Monday, then Mondays are {1st, 8th, 15th, 22nd, 29th
If 2nd Jan is a Monday, then (2nd, 9th, 16th, 23nd, 30th} are Mondays.
If 3rd Jan is a Monday, then (3rd, 10th, 17th, 24th, 31s) are Mondays.
But if 4th Jan is a Monday we can get only 4 Mondays in January.

P(in a
leap year) =

(ii) In a non-leap year also there are 31 days in January.

3
P(in a non-leap year)
=

Example 10: A carton has 500 vests of which 480 are good, 15 have minor defects and 5 have ma
with
defects. There are 2 traders A and B. A accepts only the good vests whereas B rejects the vests w
major defects. What is the probability that a vest is acceptable to (i) A and (ii) B?

Solution:
No. of good vests 480 24
For A Probability
25
=

(1) >
Total no. of vests 500
Total no. -

No. with major defects 500-5 495 99

(it) For B> Probability = Total no. of vests
500 500 100

ICSE Mathematics 10
(354)
 A s contains 6 green, 7 red and3 white balls. A ball is chosen wvithout looking into
bag contains

probability
that the ball is
ind the 22) not white (Gi) white or green
red

Sobdion No. ofred

balls 7
Pired Total n o . of balls 6+7+3 16
No. of non-white balls 6+7 13
=
white)
Pnot Total no. of balls 6+7+3 16
a

green)
=
No. ofwhite balls+green balls6+3
Pwhite o r Total no. of balls 16 16
P
probability that a card picked up
are 12 cards numbered 1 to 12. What is the
le 12: There
Exam
domly has
a p r i m en u m b e r 2) a number between 3 and 11
Go) a cube number.
asquarenumber
Salution:
-5
9 Aprime number =23,5, 7, 11)
P(prime)=
Anumberbetween 3 and 11 =
4 5, 6, 7, 8, 9, 10) =7
Pbetween 3 and 11) =

12
Asquare number {1, 4 9} =3 =

P(square no.) =

)Acube number =
[1, 8) =2
P(cubeno.)26
of length
20 cm is drawn somewhere o n a rectangular paper
mple 13: A circle with diameter o n this paper
and assuming that the cube falls on
is thrown
and width 30 cm. If a small cube will fall
Paper, what is the probability that it (ii) outside the circle.
) inside the circle
Solution: 10x10 22x1001
Area of circde r
Probability of ffalling inside the cirde =

Area of rectangle Ixb 40 x30 7 x1200 42

2-11_31
tObability of falling outside the ircle =1- 42
white and others black. Probability of getting a
Exampl There are 60 balls in a box. Some are

there?
white ball of black ball. How many
of each coloured balls are

1 getting a

Solution:
there be x white balls.
(6)-2) black balls.
Probability 355
 black ball
Probability of white ball
= x
Probability of geting a

2
5 x = 180
3 (60-a) 2x 3(60 -x) =
180 -
3x

60 2 60
X = 36 white

No. of black balls =

60 -

36 =
24

EXERCISE 21
a
1. If a dice is rolled, what is the probability of getting
ii) number 7
(i) prime number (i) multiple of 3
of getting
what is the probability
2. If two coins are tossed,
(11) at least 1 head
(11) no head
(i) 2 heads Find the probability that the drawn
of well shuffled playing cards.
at random from a pack
3. A card is drawn
card is (iii) neither a club nor a king
(ii) a multiple of 3
neither a red card nor a black king
(1)
which x are white.
18 balls out of ball?
4. A bag contains that it will be a white
what is the probability
ball is drawn
without looking, (),findx.
i) If one
a white ball is thrice that in
the bag, the probability of drawing
If 6 white balls are put in A card is now drawn
(i) more
numbers are multiples of 3 are removed.
all cards whose
5. From a pack of playing cards,
that the card drawn is
What is the probability 2011
at random.
numbered red card.
face card
(11) an even
cuDe
(i) a dimensions. If a small
paper with 30 cm x 20 cm
drawn on a rectangular that it will tall
diameter 20 cm is
6. A circle with falls on the paper what
is the probability
that it
the paper and assuming [Take T 3.14]
=

is dropped on
circle?
circle? (i) outside the
(i) inside the have 5 Tuesdays in a
that the month of February may
7. Find the probability
(it) non-leap year
i) leap year
that there are 53 Mondays in a
8. Find the probability
(i) non-leap year 1s
(i) leap year that the nuoe
are made with 3, 7, 9, a digit can be repeated, what is the probability
9. If 2 digit numbers
()greaterthan 70 (17) a prime number i) a multiple of 3
faces on the coins
tossed together, what is the probability of getting different
10. When 2 coins are

11. When 2 dice are rolled, what is the probability of getting

of 8
number on both (11) a product of 6 (t11) a sum
the
(i) the same

random from the bag. wi

12. A bag contains 5 white,
6 red and 7 green balls. One ball is drawn at
is
probability that the ball
() red (10) not green (ii) black
ball is
balls. If the probability of drawing a green ao
13. A bag contains 6 black balls and some green
a black ball. Find the number of green balls.
14. If the probability of winning a game is 0.45, what is the probability of losing the game

ICSE Mathematics 10
(356
 are rolled, what is the probability that
2 dice

the
s u m
off tthe
numbers is
atleast 10 )the product of numbers is 8
9 even
numbers

( both a r e
the p r o b a b i l i
that a
leap year has 53 sundays?
h a tis

has 200 pens

out of which 20 are defective. One pen is taken out at random. What is the
7
A
box
defective?
probability
is not
that it
playing cards, all odd numbered cards are removed. When a card is removed from the
From a pack of p
18
remaining find the probability that it is
amultiple of3 G) a face card (7) an ace
0
14entical cards are marked with numbers from 1 to 50 and well shuffed. One card is picked up from thes e t
enti
9 that the number on the card is
Find the probability
a perfect square () a perfect cube 2) a prime number ) a factor of 24

0A game of chance has a spinning arrow pivoted at the centre of a dial as shown in
the figure. When turned the arrow points to any of the twelve numbers. What is the
probability that the arrow points at
) an odd number

) a prime number
) a number more than 5
() a perfect square
21. A dice is thrown 500 times and frequency of the outcomes are noted as given below.
Outcomee 1 23 56
Frequency 75 85100 86 90 64
Find the probability of getting
)a number less than 3 (11) a prime number
A bag contains 4 green, 12 red andx yellow balls. Express in terms of x the probability of getting a yellow

Dall. When a ball is drawn, the probability of getting

a yellow ball is double that of green bal1
What is the probability of getting a red ball?
) Find the number of yellow balls. (i)
the probability of getting
When 3 coins are tossed simultaneously, what is
at least 2 tails o) at most 1 tail
) 3 tails (i) exactly 2 tails (iin)
card is drawn at random. What is the
24L a r d s 50 and well shuffled. One
are marked with numbers from 10 to
probability that it is a numberT
(1) a perfect square
divisible by 5 with of the digits 5
(in) a multiple of 3 and 4 (io) one

written on them. When a card is randomly picked up

25 Ther
Ere are 8 cards with the letters A, B, C, D, E,
A, E, O
it is a
Tom the
set, what is the probability that
(in) letter E
)vowel (i) consonant
ot the numbers on the uppermost faces of
probability that the
sum

en2 dice are rolled together, find the

the dice is
(ii) less than 6
)
4 or 5 i) more than 10
the probability that it is
cards. Find th
27. A card
is drawt wn at random froma
well-shuffled deck of52 playing
ace nor a heart
neither a n
) not a diamond (i7)
() an ace and a queen
(io) a red or a king
11) a club or a face card
Probability 357
 28. A letter is chosen at random from the letters of the word 'TRIGONOMETRY. Find the probabilit #l
that the
letter is
() a vowel (ii) a T orO (iit) neither M nor N
29. A bag contains red and yellow balls. Of these 24 are yellow balls and the remaining red. One ball
oall isis chos
chosen
at random and the probability that it is yellow is Find the number of red balls.
30. 13 a set of 2 digit numbers is
What is the probability that one number picked from

) a multiple of 2 and 7 (ii) one digit is thrice the other

31. Two coins are tossed together 100 times and the results are as follows:

No. of heads 0 1
28 52 20
Frequency
What is the probability of
i1) at most 1 head
i) one head
black. If the probability of drawing a white ball is twice
32. A bag has 24 balls. Some are white and remaining balls.
white
that of drawing a black ball, find the number of
33. If two dice are rolled, What is the probability that
() the sum of the numbers on the dice is a prime number?
of the numbers the dice is 8?
(i) the sum on

a box. If the probability of getting

a red ball is of getting a
34. There 16 red balls and s o m e white balls in
are
in the box?
white ball, how many white balls are
in a bag. A ball is drawn at random from
the bag.
35. There a r e 30 blue balls and x red balls
is red.
for the probability that the ball drawn
(i) Write down, in terms of x, expression
an

is find x.
(i) Given that the probability 13
1 coins and twenty 50 paise coins in his coin box? If he
36. A boy had thirty 2 rupee coins, twenty-five rupee that it is
what is the probability
picks upa coin from the box,
2 coin (ii) either 2 or a R1 coin (17) not a l coin
1) a

MISCELLANEOUS EXERCISE
7, what is the probability that number is
digit numbers are made with 3, 5 or a
1. If two
number (ii) a multiple of 5
(i) a prime
(ii) less than 40.
2. In an aquarium, there are 36 red and some black fish. When a fish is removed, the probability or Bet

11
black fish is Find the number of black fish.
20
t
3. ldentical cards are marked 1 to 100. When a card is drawn at random, what is the probability that
i) a multiple of 11 (i) a perfect square
(iti) a number in which one digit is twice the other (io) a multiple of 2 and 3
(o) a prime number containing the digit 9
er,
4. On a square paper of side 20 cm, a circle of diameter 10 cm is drawn. Assuming that it falls on u

what is the probability that a tiny cube dropped on the paper falls Take T-
) within the circle (ii) outside the circle?

(358 ICSE Mathematics 10

 d coins are tossed, what is the probability of getting
W h e n3 u n b i a

(0 no heads
(i) at least 2 tails
is
cards is well
is well shuffled and a card is randomly picked up, what is the probability that
deck
k of playing
Whena
6
it is
card or a queen (i) a multiple of 3
red
a
( (io) a nine and an ace
face c a r d
red
a
(it) letter is chosen randomly. What is the probability that it is a
7. From the letters
of the word MATHEMATICS, a

(0 Vowel ii) Consonant

M or T
(it)
rolled, hat is the probability that on the uppermost faces
twodice
are
&When

two numbers is less than 5 (i) the product of the numbers is 6

0 the sum of
divisible by 5
the s u m is
(it)
has the following
numbers marked on its 6 faces: 1, 2, 3, -1,-2, -3. What is the probability of getting
9 Adice
a positive integer (i) an integer less than-1

the smallest integer (iv) the smallest prime number

(it)
2014]
arrow pivoted at the centre of a dial is shown in the figure. When turned,
10. A spinning
that the 6
the arrow points to any of the 8 numbers. What is the probability arrow

points at
(i) an odd number
) a perfect cube
(ii) a factor of 8 (iv) the smallest prime number
were shuffled and put in a box. If a card is picked up froom
11. Odd numbered cards 1, 3, 5, 7, 9, 11, 27, 29
this box, what is the probability that the number on the card is a

(i) factor of 21 (ii) multiple of 5

) prime number

lot of 750 mangoes is 0.008. How many mangoes are bad in

12 The probability of getting a bad mango in a

the lot?

BOARD PAPER QUESTIONS

probability that the
Adice is thrown once. What is the
() number is even (ii) number is greater than 2? 2009]
Lards marked with numbers 1, 2, 3, 4, .20 are well shuffled and a card is drawn at random. What is the
probability that the number on the card is
iii) a perfect square? [20101
) a prime number (i) divisible by 3
the probability of drawing a black ball is two-fifths
Abox contains someblack balls and 30 white balls. If
4White ball, find the number of black balls in the box. [20131
4 is drawn at random from the bag. Find the
Dag contains 5 white balls, 6 red balls and 9 green balls. A ball
probability that the ball drawn is:
) a green ball (i) a white or a red ball

n) is neither a green ball nor a white ball. 2015]

5.
e numbers has cards marked with 11, 12, 13,..
ot 40.
A card is drawn at random. Find the Probability
at the number on the card drawn is:

)A perfect square (i1) Divisible by 7 [20161

Probability 359
 6. Sixteen cards are labelled as a, b, c, d,., m, n, o, p. They are put in a box and shuffled. A boy is as

a card from the box. What is the probability that the card drawn is:
sked to draw
() A vowel? (i) A consonant? (it) None of the letters of the word 'median',

andom 2017
7. Cards bearing numbers 2,4, 6,8, 10, 12, 14, 16, 18 and 20 are kept in a bag8 A card is drawn at r
the bag. Find the probability of getting a card which 1s;
from
() a prime number. (i1) a number divisible by 4.
(in) a number that is a multiple of 6. (io) an odd number.
8. There are 25 discs numbered 1 to 25. They are put in a closed box and shaken thoroughly. A disc 2018)
lisc isis cdrawn
at random from the box. Find the probability that the number on the disC 1;
) an odd number (ii) divisible by 2 and 3 both.
(ii) a number less than 16.
9. Each of the letters of the world 'AUTHORIZES' is written on identical circular discs and put in a bag, Th
2019
are well shuffled. If a disc is drawn at random from the bag, what is the probability that the letter is: ey
(i) A vowel
(i) One of the first 9 letters of the English alphabet which appears in the given word
(in) One of the last 9 letters of the English alphabet which appears in the given word?
2020

LET'S RECALL

1. Probability of an event or an outcome =

No. of favourable outcomes
No. of all possible outcomes
2. P(event) + P(not event) = 1
3. 0< Probability of an event 1
4. If the probability of an event = 0, the event is an impossible event.
5. If the probability of an event = 1, the event is a certain event or a sure event.
6. If a coin is tossed twice 2 coins tossed
or are
simultaneously, the total no. of possible outcomes =

2 =4
7. If a coin is tossed n times or n coins are tossed together =2" outcomes
8. If 2 dice rolled
are
possible.
are
simultaneously or one dice two times, the total no. of outcomes =6 =
36
9. (1) At least 3> 3 or more
(i) At most 2 0 or 1 or 2
10. () multiples of 2and 3 = multiples of 6 >6, 12 18, etc.
(ti) multiples of 2 or 3 multiples of 2 as well as
11. Favourable outcomes should
multiples of 32, 3, 4, 6, 8, 9, 10, 12, etc.
be listed in each case.

DO YOU KNOW?
BLAISE PASCAL (1623-1662)
Blaise Pascal, a French
mathematician, was a child prodigy. He
first digital calculator and invented the
for binomial coefficients in hydraulic press. He gave a tabular
number is the sum of the
the
expansion of (a + b". n Pascal's presentation
numbers above it. triangle, each
Probability had its origin in the games of
interested in gambling chance. In 1654,
from that collaboration problems, Blaise Pascal prompted by a friend
was born corresponded with Fermat and
helped in creating two major new the Theory of Probability. Blaise
treatise on the areas of
research. Pascal
subject of Projective He wrote
strongly influencing Modern Geometry and later on a
significant
Economics and Social Probability Theory
Science.
(360 ICSE Mathematics 10 Blaise Pascal