Probability Class 11

teachoo.
Ex 14.1, 1 CACHOO.cOM
A die is rolled. Let E be the event “die shows 4” and F be the event
“die shows even number”. Are E and F mutually exclusive?
Since die is rolled
So, S = {1, 2, 3, 4, 5, 6}
Given
E: die show 4
E= {4}
F : die show even number possible even number is 2,4 & 6
F = {2, 4, 6}

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Ex 14.1, 2
A die is thrown. Describe the following events:
(i) A: a number less than 7
If the die is thrown,
possible outcomes are 1, 2, 3, 4,5,6
S = {1, 2, 3, 4, 5, 6}.
A:anumber less then 7
A={1, 2, 3, 4, 5, 6}

Ex 14.1, 3 teachoo.com
An experiment involves rolling a pair of dice and recording the
numbers that come up. Describe the following events:
A: the sum is greater than 8, B: 2 occurs on either die
C: The sum is at least 7 and a multiple of 3.
Which pairs of these events are mutually exclusive?
Sample space when two dies are thrown
(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6),
(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6),
g =} (32, (3,2), (3, 3), (3, 4), (3, 5), 3, 6),
~ )(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6),
(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6),
(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)
A: The sum is greater than 8
Possible sum is greater than is 9, 10, 11, & 12
p= {2S)18515,4)(63)14.6)
~ ((5,5),(6,4),(5,6),(6,5),(6,6)

Ex 14.1, 4 teachoo.com
Three coins are tossed once. Let A denote the event ‘three heads
show”, B denote the event “two heads and one tail show”. C denote
the event “three tails show” and D denote the event ‘a head shows
on the first coin”. Which events are
Finding A, B, C, D
When 3 coins are tossed, possible results are
S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}
A: Three head shows

A = {HHH}

Ex 14.1, 5 teachoo.com
Three coins are tossed. Describe
(i) Two events which are mutually exclusive.
Since 3 coins are tossed , possible outcomes are
S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}
Two events which are mutually exclusive
Let A be the event getting only head
A = {HHH}
Let B be the event getting only tail
B= {TTT}

Three coins are tossed. Describe
(ii) Three events which are mutually exclusive and exhaustive.

Three coins are tossed. Describe
(iii) Two events, which are not mutually exclusive.

Three coins are tossed. Describe
(iv) Two events which are mutually exclusive but not exhaustive.

Three coins are tossed. Describe
(v) Three events which are mutually exclusive but not exhaustive.

Ex 14.1, 6 teachoo.com
Two dice are thrown. The events A, B and C are as follows:
A: getting an even number on the first die.
B: getting an odd number on the first die.
C: getting the sum of the numbers on the dice <5
Describe the events
If 2 dies are thrown then possible outcomes
are 1, 2, 3, 4, 5, 6 on both dies

(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6),

(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6),

g =) 3 0, B, 2), (3, 3), (3, 4), (3, 5), (3, 6),
~ (4, 0), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6),
(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6),
(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)

Ex 14.1, 7 teachoo.com
Refer to question 6 above,
State true or false: (give reason for your answer)
(i) A and B are mutually exclusive
From 16.2 ,6

(2, 1), (2, 2), (2, 3), (2, 4), (2,5), @, 6),
A=4 (4,1), (4,2), (, 3), (4,4), (4,5), 4, 6),

(6, 1), (6, 2), (6, 3), (6, 4), (6,5), (6 6)

(1, 1), (1, 2), 1, 3), G1, 4), (1, 5), (1,6),
B= 4 (3,1), (3,2), G, 3), G, 4), (3, 5), G, 6)

(5, 1), 6, 2), G, 3), G, 4), (5, 5), G, 6)
ANB=o

Ex 14.2, 1 teachoo.com
Which of the following can not be valid assignment of probabilities
for outcomes of sample space S = { 01, Wz, W3, W4, Ws, We, W7,}
°) BPR
0.1 0.01 0.05 0.03 0.01 0.2 0.6

S= { Wy, Wo, 3, W4, Ws, We, Oz}
Sum of probability

= W1 + Wz + Wz +W4+ Ws + We + 7

= 0.1+0.01+0.05 + 0.03 +0.01+0.2+0.6

= 1.00
-. Assignment of probability is valid

Ex 14.2, 2 teachoo.com
A coin is tossed twice, what is the probability that at least one tail
occurs?
When 2 coins are tossed,
Sample Space = S = {HH, HT, TH, HT}
n(S)=4
Let A be the event that at least 1 tail occurs
Hence A = {HT, TH, TT}
n(A) =3
P(A)= Number of outcomes favourable to A
( ) ~ Total number of possible outcomes
=n)
~ n(s)
3
“4

Ex 14.2, 3 teachoo.com
A die is thrown, find the probability of following events:
(i) A prime number will appear,
When a die is thrown,
Sample space = S = {1, 2, 3, 4, 5, 6}
« n(S) = 6
Prime numbers between 1 to 6 are
2,3 and5
Let A be the event of that prime number appear
« n(A)=3

A number greater than or equal to 3 will appear,

A number more than 6 will appear,

A number more than 6 will appear,

A number less than 6 will appear.

Ex 14.2, 4 teachoo.com
A card is selected from a pack of 52 cards.

(a) How many points are there in the sample space?

Since these are 52 cards

these are 52 points in this sample space

n(S} = 52

Ex 14.2,5 teachoo.com
A fair coin with 1 marked on one face and 6 on the other and a fair
die are both tossed. Find the probability that the sum of numbers
that turn up is
(i) 3 Result | Result
of coin| of die
1 1 2
If the coin is tossed we get only 1 or 6 1 2 3
If a die is thrown we get 1, 2, 3, 4, 5, 6 1 3 4
1 4 5
1 5 6
Hence, 1 6 7
6 1 7
$= {i 4), (1, 2), (1, 3), (1, 4), (1, 5), (1, at 6 2 8
(6, 1), (6, 2), (6, 3), (6, 4), (6,5), (6, 6)) g 3 8
6 4 10
6 5 11
n(S)=12 6 6 PI

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Ex 14.2, 6 "
There are four men and six women on the city council. If one council
member is selected for a committee at random, how likely is it that
it is a woman?
Total members = 10
Hence

n(S) = 10
Total women members =6
Let A be the event that women is selected
Hence n(A) =6

Ex 14.2, 7 teachoo.com
A fair coin is tossed four times, and a person win Re 1 for each head
and lose Rs 1.50 for each tail that turns up. From the sample space
calculate how many different amounts of money you can have after
four tosses and the probability of having each of these amounts.
If a coin is tossed 4 times the following results are possible
Profit on Heads Loss on tails Amount of
money
4 head 0 tail 4x1=4 0x1.5=0 4-0=4
3 head 1 tail 3x1-+3 1x15=1.5 3-1.5=1.5
2 head 2 tail 2x1=2 2x1.5=3 2-3=-1
1 head 3 tail 1x1=1 3x15=45 1-45=-3.5
0 head 4 tail Ox1=0 4x15=6 0-6=-6
Hence, Different amount of money = {4, 1.5, —1, -3.5, —6}

Ex 14.2, 8 teachoo.com
Three coins are tossed once. Find the probability of getting

(i) 3 heads

If 3 coins are tossed various combination possible are

S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}

n(S) = 22= 8

Let A be the event of getting 3 head

A= {HHH}

on{A)=1

Probability of 3 heads = P(A)

Ex 14.2, 9 teachoo.com
f= is the probability of an event, what is the probability of the
event ‘not A’
Let A be the event given
2
So, P(A) = —
Probability of not A
= P(A’)
=1-P(A)
-1-2
11
_ 11-2
"41
=2
“44

Ex 14.2, 10 teachoo.com
A letter is chosen at random from the word ‘ASSASSINATION’. Find
the probability that letter is (i) a vowel
Letter in the word ASSASSINATION
Number of total letters in the given word = 13
n(S) = 13
Let A be the event getting vowels
So, A={A, A, A, |, |, O}
n(A)=6
Probability of getting vowels = P(A)
_ Number of vowels in word
~ Total number of letters in word
= nA)
~ n(S)
-&
“43

Ex 14.2, 11 teachoo.com
In a lottery, person choses six different natural numbers at random
from 1 to 20, and if these six numbers match with the six numbers
already fixed by the lottery committee, he wins the prize. What is
the probability of winning the prize in the game? [Hint: order of the
numbers is not important.]
These are 20 natural number from 1 to 20
Total numbers = 20
Number to be chosen=6
Number of ways choosing 6 natural number from 1 to 20 = 7°C,

_ 20! 20! © 38760

~ 6120-6)! 6114! ways

Ex 14.2, 12 teachoo.com
Check whether the following probabilities P(A) and P(B) are
consistently defined
(i) P(A)= 0.5, P(B) = 0.7, P(A n B) =0.6
P(A) & P(B) are consistently defined if
1. P(AMB)<P(A) & P(A 9 B) < P(B)
2. P(AUB)> P(A) & P(A U B) > P(B)
Given
P(A) = 0.5, P(B) = 0.7, P(A A B) = 0.6
Here,
P(A 1 B) > P(A).
Hence, P(A) and P(B) are not consistently defined.

Check whether the following probabilities P(A) and P(B) are consistently defined
P(A) = 0.5, P(B) = 0.7, P(A ∩ B) = 0.6

Ex 14.2, 13 teachoo.com
Fill in the blanks in following table:
P(A) P(B) P(A B) P(AUB)
. 1 1 1
MG 3 rr
We know that
P(A U B) = P(A) + P(B) — P(A 0 B)
Putting values
1 1 1
P(AUB)=7 +2-—
543-1
7
Hence P(A UB) =——
lence P(. )=75

Ex 14.2, 14 teachoo.com
Given P(A) == and P(B) = = Find P(A or B), if Aand B are mutually
exclusive events.

3 1
P(A)=— , P(B)=—
Since A and B are mutually excusive
So, P(AN B)=0
Now,
P(A or B) = P(A U B)
We know that

P(A U B) = P(A) + P(B) — P(A 2 B)

Ex 14.2, 15 teachoo.com
If E & F are events such that P(E) = : , P(F)= ; and P(E and F) = er find:
(i) P(E or F)
P(E and F) = P(E 9 F) =—
We need to find
P(E or F) = P(E U F)
We know that
P(E U F) = P(E) + P(F)— P(E 9 F)
Putting values
11 1
P(EUF)=—+- --
4 2 8

Ex 14.2, 16 teachoo.com
Events E and F are such that P(not E or not F) = 0.25, State whether
E and F are mutually exclusive.
Given that
P (not E or not F) = 0.25 Demorgan’s law
P(E’ UF’) = 0.25 If (A’N BY) =(A U By’
P (ENF) =0.25 or (A’ U B’) =(A 1 By’
1-P(EMF)=0.25 (By Demorgan law)
1-0.25 =P (ENF)
0.75 = P(E MF)
P (ENF) =0.75
Since P (ENF) #0

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Ex 14.2, 17
A and B are events such that P(A) = 0.42, P(B) = 0.48 and P(A and B)
= 0.16. Determine
(i) P(not A),
P(A) = 0.42
P(not A)

=1- P(A)

=1-0.42

= 0.58

Ex 14.2, 18 teachoo.com
In Class Xl of a school 40% of the students study Mathematics and
30% study Biology. 10% of the class study both Mathematics and
Biology. If a student is selected at random from the class, find the
probability that he will be studying Mathematics or Biology.
Let B be the event of student studying biology.
& M be the event of student studying maths
Hence
Probability student study maths = 40%
40
P(M)= 40% =— > =0.4
Probability student study biology = 30%
30
P(B) = 30% => =0.3

Ex 14.2, 19 teachoo.com
In an entrance test that is graded on the basis of two examinations,
the probability of a randomly chosen student passing the first
examination is 0.8 and the probability of passing the second
examination is 0.7. The probability of passing at least one of them
is 0.95. What is the probability of passing both?
Let A be the event of passing the first examinations

& B be the event of passing second examination
Given
Probability of passing first exam = P(A) = 0.8
Probability of passing second exam = P(B) = 0.7
Given that
Probability of passing at least 1 of them = 0.95
Hence, P(A U B) = 0.95

Ex 14.2, 20 teachoo.com
The probability that a student will pass the final examination in
both English and Hindi is 0.5 and the probability of passing neither
is 0.1. If the probability of passing the English examination is 0.75,
what is the probability of passing the Hindi examination?
Let E be the event of that a student passes in English
& H be the event that a student passes in Hindi

Given,
Probability of passing both English & Hindi = 0.5

P(EQ H)=0.5
Probability of passing in neither subject = 0.1

P(E’N H’) = 0.1

Ex 14.2, 21 teachoo.com
In a class of 60 students, 30 opted for NCC, 32 opted for NSS and 24
opted for both NCC and NSS. If one of these studentsis selected at
random, find the probability that
(i) The student opted for NCC or NSS.
There are 60 studentin a class
n(S) = 60
Let A be the event that a student opted for NCC
& B be the event that a student opted for NSS
Given that
30 students opted for NCC
So, n(A) = 30

Example 1 teachoo.com
Consider the experiment of rolling a die. Let A be the event ‘getting
a prime number’, B be the event ‘getting an odd number’. Write the
sets representing the events

(i) AorB

While rolling a die we can get 1, 2, 3, 4,5, 6

So, S = {1, 2, 3, 4, 5, 6},

Prime number between 1 to 6 are 2, 3 and5

So, A= {2, 3, 5}

Odd number between 1 to6 are 1,3 and5

So, B = {1, 3, 5}

Example 2 teachoo.com
Two dice are thrown and the sum of the numbers which come up
on the dice is noted. Let us consider the following events associated
with this experiment
A: ‘the sum is even’.
B: ‘the sum is a multiple of 3’.
C: ‘the sum is less than 4’.
D: ‘the sum is greater than 11’.
Which pairs of these events are mutually exclusive?
If two dice are thrown then possible
outcomesare 1, 2, 3, 4, 5 & 6 on both dies
Hence
(1,1), (1, 2), (1,3), (1, 4), C1, 5), (1, 6)
(2, 1), (2, 2), (2,3), (2, 4), (2, 5), (2,6)
$= E 1), 3,2), B,3),G, 4), 3,5), GB, ;
(4, 1), (4, 2), (4,3), (4, 4), (4, 5), (4,6)
(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6)
(6, 1), (6, 2), (6, 3), (6,4), (6,5), (6 6),

teachoo.com
Example 3
A coin is tossed three times, consider the following events.
A: ‘No head appears’, B: ‘Exactly one head appears’ and C: ‘At least
two heads appear’. Do they form a set of mutually exclusive and
exhaustive events?
If 3 coins are tossed , possible outcomes are
S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}
A: no head appear
Hence only tail appear in all 3 times
So A={TTT}

Example 4 teachoo.com
Let a sample space be S = {w,, Wy,..., We}.Which of the following
assignments of probabilities to each outcome are valid?
Outcomes W, W, Wz Wy Ws We
1 1 1 1 1 1
6 6 6 6 6
S= { 1, Wo, Wz, W4, Ws, We}
Sum of probability
=, + Wz + Wz + W4 t We + We
14114141
a
6 6 66 66
=°
“6
=1
«. Assignment of probability is valid

Example 5 teachoo.com
One card is drawn from a well shuffled deck of 52 cards. If each
outcome is equally likely, calculate the probability that card will be
(i) a diamond
5432
weee: A
ss 2 a a4 + ¢ +
Since there 52 cards > a
"
a
n(S) = Total number 9
of cards = 52
s
v
There are 13 diamond cards
Let A be event that diamond card is withdrawn
So, n(A) = 13

Example 6 teackoo.com
A bag contains 9 discs of which 4 are red, 3 are blue and 2 are yellow.
The discs are similar in shape and size. A disc is drawn at random
from the bag. Calculate the probability that it will be
(i) red,
There are 9 discs of which
4are red
3 are blue
& 2 are yellow
Total number of possible outcomes is 9
n(sj=9
Let A be the event that the disc is red
B be the event that the disc is yellow
C be the event that the disc is blue

Example 7 teachoo.com
Two students Anil and Ashima appeared in an examination. The
probability that Anil will qualify the examination is 0.05 and that
Ashima will qualify the examination is 0.10. The probability that
both will qualify the examination is 0.02. Find the probability that
(a)Both Anil and Ashima will not qualify the examination.
Let E be the event that Anil will qualify the examination
& F be the event that Ashima will qualify the examination

Given

Probability that Anil will qualify the exam = P(E) = 0.05

Probability that Ashima will qualify the exam = P(F) = 0.10

Example 8 teachoo.com
A committee of two persons is selected from two men and two
women. What is the probability that the committee will have
(a) no man?
If no man is selected, it means only women are selected
So, we have to select 2 women
Total number of persons =2+2=4
Number of persons to be selected = 2
P(no man is selected) - Number of ways both women is selected (1)
Total number of ways
Total Number of ways = *C,
— ab 4h 4x BX 2g
24-2)! 22) 2x 1x 2

Example 9 teackoo.com
On her vacations Veena visits four cities (A, B, C and D) in arandom
order. What is the probability that she visits
(i) A before B?
4 cities can be visited in any of following order

ABCD, ABDC, ACBD, ACDB, ADBC, ADCB,
S= BACD, BADC, BDAC, BDCA, BCAD, BCDA,

~ ) CABD, CADB, CBDA, CBAD, CDAB, CDBA,

DABC, DACB, DBCA, DBAC, DCAB, DCBA
n(S) = 24
Let E be the event that “she visits A before B“

ABCD, ABDC, ADBC, ACDB, ADBC, ADCB,
Hence, E= | CABD, CADB, CDAB,
DABC, DACB, DCAB,

Example 10 teachoo.com
Find the probability that when a hand of 7 cards is drawn from a well
shuffled deck of 52 cards, it contains
{i) all Kings
7 cards are to be chosen from 52 cards
Total number of combinations (hands) possible = 52C,

_ 82th

7\(52-7)! 7145!

Let A be the event that all kings are selected
There are only 4 kings in a pack of 52 cards

Example 11 teachoo.com
If A, B, C are three events associated with a random experiment,
prove that P(A U B U C) = P(A) + P(B) + P(C) - P(An B) - P(An C)
-P(BNC)+P(ANBNC)
Let BUC=E
So, P(AUBUC)=P(AU E)
= P(A) + P(E)— P(A o E)
= P(A) + P(E)- P(A (BU C))
=P(A)+P(E)—P(ANB)U(ANC)) (1)
We find P(E) & P((A 9 B) U (A C)) separately
Finding P(E)
P(E)=P(BUC)
= P(B) + P(C) - P(Bn C) (2)

Example 12 teachoo.com
In a relay race there are five teams A, B, C, D and E.
(a) What is the probability that A, B and C finish first, second and
third, respectively.
3 teams will win from 5 teams
Hence
Total number of orders = °P.,
5!
~ (5-3)!
5!
“21
_5X4X3 x2!
~ 2!
=60

teachoo.com
Example 1
Two coins (a one rupee coin and a two rupee coin) are tossed
once. Find a sample space.
Both coins are same,
either head comes or tail comes
Possible combination
when 2 coins are tossed
We denote head by H
a tall byT | coin 1 | coin 2 |
H H
H T
T H
Hence the sample space is T T
S = {(H,H), (H,T), (TH), (T.T)}

Example 2 teackoo.com
Find the sample space associated with the experiment of rolling a
pair of dice (one is blue and the other red) once.
Also, find the number of elements of this sample space.
Possible results on both dies are
1, 2, 3, 4, 5, 6 only
Hence sample size is
(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6),
(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6),
5 -] BD, (3, 2), 8, 3), 3, 4), (3,5), B, 6),
~ ) (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6),
(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6),
(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)
Number of elements in sample space = 6 x 6
=36

Example 3 teackoo.com
In each of the following experiments specify appropriate sample space
(i) A boy has a 1 rupee coin, a 2 rupee coin and a 5 rupee coin in his
pocket. He takes out two coins out of his pocket, one after the
other.
Let Q denote a 1 rupee coin,
H denotes a 2 rupee coin Q H
& R denotes a 5 rupee coin. Q R
H Q
He takes out two coins, one after the other H R
R Q
All possible outcomes are R H
S = {QH, QR, HQ, HR, RQ, RH}

Example 3
In each of following experiments specify appropriate sample space
(ii) A person is noting down the number of accidents along a busy highway during a year

Example 4 teackoo.com
A coin is tossed. If it shows head, we draw a ball from a bag
consisting of 3 blue and 4 white balls; if it shows tail we throw a
die. Describe the sample space of this experiment.

The results on tossing coin

are heads and tails

Let heads be denoted by H

& tails by T

Also, 3 blue & 4 white balls

be denoted by B,, B,, B, and W,, W,, W; and W,

Results of die are 1, 2,3, 4,5, 6

Example 5 teachoo.com
Consider the experiment in which a coin is tossed repeatedly until
a head comes up. Describe the sample space.
Let Head be H
Coin tossed
& Tail be T
‘time | tine | time | time time | - i
coin tossed
H - - - -
T H - - - Head Tail
T T H 7 - coin tossed
T T T H - ——
Head Tail
T T T T H
S = {H, TH, TTH, TTTH, TTTTH....}

Misc 1 teachoo.com
A box contains 10 red marbles, 20 blue marbles and 30 green marbles.
5 marbles are drawn from the box, what is the probability that
(i) all will be blue?
5 marbles are to be chosen from 60 (10 red + 20 blue + 30 green)
Hence
n(S) = °C,
Let A be the event that all are blue
Hence out of 20 blue marbles 5 will be chosen
n(A) = 27°C,
20

= m4) __€s

P(A)= ns) ¢,

Misc 2 teachoo.com
4 cards are drawn from a well-shuffled deck of 52 cards. What is the
probability of obtaining 3 diamonds and one spade?

Total 4 cards are to be selected out of 52 cards

Hence

Total ways to select 4 cards= °C,

We need to select 3 diamond card & 1 spade card

Total number of diamond card = 13

Number of diamond card to be selected = 3

Total ways to select 3 diamond card = °C,

Misc 3 teachoo.com
A die has two faces each with number ‘1’, three faces each with
number ‘2’ and one face with number ‘3’. If die is rolled once,
determine
(i) P(2)
A normal die has 6 faces 1, 2, 3, 4, 5, 6
But in this question
die has following 6 faces
1, 1,2, 2, 2,3
n(S)=6
We need to find P(2)
There are 3 three 2’s

Misc 4 teachoo.com
In a certain lottery 10,000 tickets are sold and ten equal prizes are
awarded. What is the probability of not getting a prize if you buy
(a) one ticket
Since, 1 ticket is chosen out of 10000 tickets
n(S) = 10000¢ |
10000! 10000 x 9999!
=———_ = ———— = 10000
119999! 1x 9999!
Now out of 10000 tickets only 10 have a prize
Hence number of tickets not having prize
= 10000 — 10
= 9990

Misc 5 (a) Method 1 teachoo.com
Out of 100 students, two sections of 40 and 60 are formed. If you and
your friend are among the 100 students, what is the probability that
(a) you both enter the same sections?
Both enter same section
Both enter Section A Both enter Section B
Pott [Friends [other [La Total Fonds |Other
All All
100 2 98 100 2 98
students students
Studentin Studentin
40 2 38 . 60 2 58
Section A Section B
n{S) = *°°Cag n(S) = Ceo
Let A be event that both Let B be event that both
enter sectionA enter section B
n{A) = C35 n(B) = Cog

Misc 6 teachoo.com
Three letters are dictated to three persons and an envelope is
addressed to each of them, the letters are inserted into the envelopes
at random so that each envelope contains exactly one letter. Find the
probability that at least one letter is in its proper envelope.
Let the envelope be denoted by A, B, C
and the corresponding letters are a, b, c
The letters are inserted into the envelopes at random,
& each envelope contain exactly one letter
Possible combinations can be

Aa Bb Cc

Aa Be Cb

Ab Ba Cc

Ab Bc Ca

Ac Bb Ca

Ac Ba Cb
So, Total number of possible cases = 6

Misc7 teachoo.com
A and B are two events such that P(A) = 0.54, P(B) = 0.69 and
P(A B) = 0.35. Find
(i) P(A U B)
Given
P(A) = 0.54, P(B) = 0.69, P(A N B) = 0.35
We know that
P (AU B) = P(A) + P(B) - P(A 0 B)
Putting values
= 0.54 + 0.69 - 0.35
= 0.88

Misc 7
A and B are two events such that P(A) = 0.54, P(B) = 0.69 and P(A ∩ B) = 0.35. Find
(ii) P(A′ ∩ B′)

Misc 7
A and B are two events such that P(A) = 0.54, P(B) = 0.69 and P(A ∩ B) = 0.35. Find
(iii) P(A ∩ B′)

Misc 7
A and B are two events such that P(A) = 0.54, P(B) = 0.69 and P(A ∩ B) = 0.35. Find
(iv) P(B ∩ A′)

Misc 8 teachoo.com
From the employees of a company, 5 persons are selected to
represent them in the managing committee of the company.
Particulars of five persons are as follows:
S.No Name Sex Age in years

1. Harish M 30

2. Rohan M 33

3. Sheetal F 46

4. Alis F 28

5. Salim M 41
A person is selected at random from this group to act as a
spokesperson. What is the probability that the spokesperson will
be either male or over 35 years?
Total number of persons = 5
So, n(S)=5

teachoo.com
Misc 9
If 4-digit numbers greater than 5,000 are randomly formed from
the digits 0, 1, 3, 5, and 7, what is the probability of forming a
number divisible by 5 when,
(i) the digits are repeated?
Digit number greater than 5000 can
be formed with either 5 in beginning
or 7 in beginning
5
or
7

teachoo.com

Misc 10
The number lock of a suitcase has 4 wheels, each labelled with ten
digits i.e., from 0 to 9. The lock opens with a sequence of four digits
with no repeats. What is the probability of a person getting the
right sequence to open the suitcase?
There are 10 digits out of which 4 digits to be chosen with no
repeats
Hence,
n(S) = Total 4 digit numbers out 10 digits

=10p, {We used permutation as order is important)

10! 10!_ 10x 9x 8x 7x 6!
“(o-4)! 6! 6! = 5040

Ex 16.1, 1 teachoo.com
Describe the sample space for the indicated experiment:
A coin is tossed three times.
When a coin is tossed,
we get either heads or tails
H H H
Let heads be denoted by H H H T
and tails cab be denoted by T H T H
T H H
Hence the sample space is
T T H
S = {HHH,
H T T
HHT, HTH, THH,
T H T
TTH, HTT, THT,
T T T
TIT}

Ex 16.1, 2 teachoo.com
Describe the sample space for the indicated experiment: A die is
thrown two times.
When a die is thrown,
either of 1, 2, 3, 4, 5, 6 is possible
The results of 2 Dies can be as follows
(1, 1), (1, 2), (1, 3), (2, 4), (1, 5), (1, 6),
(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6),
5 = |B 1 (3,2), 8,3), (3, 4), (3,5), (3, 6),
~ ) (4,1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6),
(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6),
(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)

Ex 16.1, 3 teachoo.com
Describe the sample space for the indicated experiment: A coin is
H H H H
When a coin is tossed, H H H T
H H T H
we get either heads or tails H H T T
H T H H
Let head d d by H H 7 4 T
et head denoted by H T T H
& tail denoted by T H T T T
T H H H
T H H T
T H T H
Hence, T H T T
T T H H
T T H T
HHHH, HHHT, HHTH, HHTT, T T T H
Ss HTHH, HTHT,HTTH,HTTT, T T T T
~ ) THHH, THHT, THTH, THTT, J
TTHH, TTHT, TTTH, TTTT

teachoo.com
Ex 16.1, 4
Describe the sample space for the indicated experiment: A coin is
tossed and a die is thrown.
When a coin is tossed either we can get heads or tails
Let heads be denoted by H
and tails by T
When coin| When die
tossed rolled

Similarly when a die is rolled H 1
the possible results are T 2
1, 2, 3,4, 5,6 3

4

5

6

teachoo.com
Ex 16.1, 5
Describe the sample space for the indicated experiment: A coin is
tossed and then a die is rolled only in case a head is shown on the coi
When a coin is tossed
When a coin is tossed ,
either we get head or tail H T
|
Lets head be denoted by H Die thrown
& tail be denoted by T |
H1
H2
According to a question H3
When head is shown then a die is rolled ie
H6
Hence,
Total number of sample space S$ associated with the experiment
S = {H1, H2, H3, H4, H5, H6, T}

Ex 16.1, 6 teachoo.com
2 boys and 2 girls are in Room X, and 1 boy and 3 girls in Room Y.
Specify the sample space for the experiment in which a room is
selected and then a person.
Room X Room Y
Let B,, B, and G,, G, are the B,, and G;, G,, G, are the
boys & girls in room X boys & girls in room Y
Possible outcomes Possible outcomes
When room X is selected then When room Y is selected then a
a person person
{XB,, XB,, XG,, XG,} {YB3, YG3, YG,, YG5}

Hence

The sample space S is

S = {XB,, XB,, XG,, XG,, YB, YG, YG,, YG,}

Ex 16.1, 7 teachoo.com
One die of red colour, one of white colour and one of blue colour are
placed in a bag. One die is selected at random and rolled, its colour an
the number on its uppermost face is noted. Describe the sample space
Let R denote red die
W denote white die
& B denote blue die
When a die is rolled,
possible outcomes are 1, 2, 3, 4,5, 6
Possible outcome when red die is selected
{R1, R2, R3, R4, R5, R6}
Possible outcomes when white die is selected
{W1, W2, W3, W4, WS, W6}

Ex 16.1, 8 teachoo.com
An experiment consists of recording boy-girl composition of families
with 2 children.
(i) What is the sample space if we are interested in knowing
whether it is a boy or girl in the order of their births?
A family with 2 child may be | ar child 2m child |
either girl or boy
B B
B G
Let G denote girl child
G B
& B denote boy child
G G
S = {BB, BG, GB, GG}

What is the sample space if we are interested in the number of girls in the family?

teachoo.com
Ex 16.1, 9
A box contains 1 red and 3 identical white balls. Two balls are
drawn at random in succession without replacement. Write the
sample space for this experiment.
A box contain 1 red & 3 identical white balls
Let R denote the red ball
& W denote the identical white ball
The two balls selected at random in succession without
replacement | 2% ball | 2% ball
R Ww

Th | i

e sample space ais w R
S = {RW, WR, WW} Ww Ww

Ex 16.1, 10 teachoocom
An experiment consists of tossing a coin and then throwing it second
time if a head occurs. If a tail occurs on the first toss, then a die is
rolled once. Find the sample space.

If a coin tossed there can be either head or tail

Let head denote by H

& tail denote by T

Similarly

If a die is rolled possible outcomes

are 1, 2,3,4,5,&6

Ex 16.1, 11 teachoo.com
Suppose 3 bulbs are selected at random from a lot. Each bulb is
tested and classified as defective (D) or non-defective (N). Write
the sample space of this experiment?
2 bub] 3b

3 bulbs are to be selected at random D D D
Each bulb is tested D D N
D - denote for defective D N D
& N - denote for non-defective D N N

N D D

N D N

N N D

N N N
Sis the sample space of this experiment
S = {DDD, DDN, DND, DNN, NDD, NDN, NND, NNN}

teachoo.com

Ex 16.1, 12
A coin is tossed. If the outcome is a head, a die is thrown. If the die
shows up an even number, the die is thrown again. What is the
sample space for the experiment?
When a coin is tossed, either head comes as tail
Let denote head by H

& Tail by T
Similarly when a die is rolled,
the possible results are
1,2,3,4,5&6

Ex 16.1, 13 teachoo.com
The numbers 1, 2, 3 and 4 are written separately on four slips of
paper. The slips are put in a box and mixed thoroughly. A person
draws two slips from the box, one after the other, without
replacement. Describe the sample space for the experiment.
Four slips marked as 1, 2, 3, 4, are in the box
The slips are put in the box & mixed
A person draws two slips one after other without replacement
that means (1, 1}, (2, 2}, (3, 3) & (4, 4} are not possible
S be the sample space with the experiment

(1, 2), (1, 3), (1, 4),
ga) (2 Ms(2, 3), (2,4),

~ ) (3,4), (3, 2), (3, 4),
(4, 1), (4, 2), (4, 3)

Ex 16.1, 14 teachoo.com
An experiment consists of rolling a die and then tossing a coin once
if the number on the die is even. If the number on the die is odd, the}
coin is tossed twice. Write the sample space for this experiment.

If a die is rolled

Possible outcomes are

1,2,3,4,5&6

Similarly if a coin is tossed

There can be either Head or tail

Let H denotes head

& T denotes tail

teachoo.com
Ex 16.1, 15
A coin is tossed. If it shows a tail, we draw a ball from a box which
contains 2 red and 3 black balls. If it shows head, we throw a die.
Find the sample space for this experiment.
tf a coin is tossed there can be either head or tail
Let head be denoted by H
& tail be denoted by T
R, & R, be denoted by red ball
& B,, B,, and B, be denoted by black ball
If a die is thrown, the possible outcomes are
1,2,3,4,5,6

Ex 16.1, 16 teachoo.com
A die is thrown repeatedly until a six comes up. What is the sample
space for this experiment?
If a die is throw possible outcomes are
Die rolled
1, 2,3,4,5,6
6 Not 6
6 Die rolled
Not 6 6 - 6 Not 6
Not 6 Not 6 6 |
Die rolled
6 Not 6