NCERT Solutions for Class 11 Maths Chapter 14 Exercise 14.1 - Probability

NCERT Solutions for Class 11 Maths Chapter 14 Exercise 14.1 - Probability

Komal MiglaniUpdated on 07 May 2025, 02:24 PM IST

Many situations in our real life involve outcomes that cannot be predicted with certainty, such as the selection of a ball from different coloured balls, the selection of a card from a deck, the roll of a die, and the toss of a coin, etc. Here, probability comes into action. Basically, probability deals with the study of uncertain events. It provides a mathematical framework to quantify the chances of different outcomes. In this exercise, you learn about some basic concepts related to probability, like Events in probability, Types of events, such as Impossible and sure events, simple and compound events, complementary events, mutually exclusive events, and Exhaustive events. Read the NCERT to learn more about probability.

This Story also Contains

  1. NCERT Solutions for Class 11 Maths Chapter 14 Probability Exercise 14.1
  2. Topics covered in Chapter 14 Probability Exercise 14.1
  3. NCERT Solutions of Class 11 Subject-Wise
  4. Subject-Wise NCERT Exemplar Solutions

The NCERT solutions of Chapter 14 Probability Exercise 14.1 are designed by our experienced subject experts to offer a systematic and structured approach to these important concepts and help students to prepare well for exams and to gain knowledge about all the natural processes happening around them by a series of solved questions. These NCERT Solutions follow the CBSE pattern and also provide a valuable resource to the students to enhance their performance in board exams as well as competitive exams like JEE.

NCERT Solutions for Class 11 Maths Chapter 14 Probability Exercise 14.1

Question:1. 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?

Answer:

When a die is rolled, the sample space of possible outcomes:

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

Now,

E = event that the die shows 4 = {4}

F = event that the die shows even number = {2, 4, 6}

E $\cap$ F = {4} $\cap$ {2, 4, 6}

= {4} $\neq \phi$

Hence E and F are not mutually exclusive event.

Question:2(i) A die is thrown. Describe the following events:

A: a number less than 7

Answer:

When a die is rolled, the sample space of possible outcomes:

S = {1, 2, 3, 4, 5, 6} or {x : x $\in$ N, x<7}

Given, A : a number less than 7

As every number on a die is less than 7

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

Question:2(ii) A die is thrown. Describe the following events:

B: a number greater than 7

Answer:

When a die is rolled, the sample space of possible outcomes:

S = {1, 2, 3, 4, 5, 6} or {x : x $\in$ N, x<7}

Given, B: a number greater than 7

As no number on the die is greater than 7

B = $\pi{100} \phi$

Question:2(iii) A die is thrown. Describe the following events:

C: a multiple of 3.

Answer:

When a die is rolled, the sample space of possible outcomes:

S = {1, 2, 3, 4, 5, 6} or {x : x $\in$ N, x<7}

Given, C : a multiple of 3

C = {3, 6}

Question:2(iv) A die is thrown. Describe the following events:

D: a number less than 4

Answer:

When a die is rolled, the sample space of possible outcomes:

S = {1, 2, 3, 4, 5, 6} or {x : x $\in$ N, x<7}

Given, D : a number less than 4

D = {1, 2, 3}

Question:2(v) A die is thrown. Describe the following events:

E: an even multiple greater than 4

Answer:

When a die is rolled, the sample space of possible outcomes:

S = {1, 2, 3, 4, 5, 6} or {x : x $\in$ N, x<7}

Given, E : an even number greater than 4

S1 = Subset of S containing even numbers = {2,4,6}

Therefore , E = {6}

Question:2(vi). A die is thrown. Describe the following events:

F: a number not less than 3

Answer:

When a die is rolled, the sample space of possible outcomes:

S = {1, 2, 3, 4, 5, 6} or {x : x $\in$ N, x<7}

Given, F : a number not less than 3

F = {x: x $\in$ S, x $\geq$ 3 } = {3, 4, 5, 6}

Question:2.(vi) A die is thrown. Describe the following events:

Also find (a) $A\cup B$

Answer:

A = {1, 2, 3, 4, 5, 6}

B= $\phi$

$\therefore$ A $\cup$ B = {1, 2, 3, 4, 5, 6} $\cup$$\phi$ = {1, 2, 3, 4, 5, 6}

Question:2.(vi) A die is thrown. Describe the following events:

Also find (b) $A\cap B$.

Answer:

A = {1, 2, 3, 4, 5, 6}

B= $\phi$

$\therefore$ A $\pi{80} \cap$ B = {1, 2, 3, 4, 5, 6} $\pi{80} \cap$$\phi$ = $\phi$

Question:2.(vi) A die is thrown. Describe the following events:

Also find (c) $B\cup C$

Answer:

B= $\phi$

C= {3, 6}

$\therefore$ B $\cup$ C = $\phi$$\cup$ {3, 6} = {3, 6}

Question:2.(vi) A die is thrown. Describe the following events:

(d) Also find $E\cap F$

Answer:

E = {6}

F = {3, 4, 5, 6}

$\therefore$ E $\pi{80} \cap$ F = {6} $\pi{80} \cap$ {3, 4, 5, 6} = {6}

Question:2.(vi) A die is thrown. Describe the following events:

Also find (e) $D\cap E$

Answer:

D = {1, 2, 3}

E = {6}

$\therefore$ D $\pi{80} \cap$ E = {1, 2, 3} $\pi{80} \cap$ {6} = $\phi$ (As nothing is common in these sets)

Question:2.(vi) A die is thrown. Describe the following events:

Also find (f) $A-C$

Answer:

A = {1, 2, 3, 4, 5, 6}

C = {3, 6}

$\therefore$ A - C = {1, 2, 3, 4, 5, 6} - {3, 6} = {1, 2, 4, 5}

Question:2.(vi) A die is thrown. Describe the following events:

Also find (g) $D-E$

Answer:

D = {1, 2, 3}

E = {6}

$\therefore$ D - E = {1, 2, 3} - {6} = {1, 2, 3}

Question:2.(vi) A die is thrown. Describe the following events:

Also find (h) $E\cap F'$

Answer:

E = {6}

F = {3, 4, 5, 6}

$\therefore$ F' = {3, 4, 5, 6}' = S - F = {1, 2}

$\therefore$ E $\pi{80} \cap$ F' = {6} $\pi{80} \cap$ {1, 2} = $\phi$

Question:2.(vi) A die is thrown. Describe the following events:

Also find (i) ${F}'$

Answer:

F = {3, 4, 5, 6}

$\therefore$ F' = {3, 4, 5, 6}' = S - F = {1, 2}

Question:3(a) An experiment involves rolling a pair of dice and recording the numbers that come up. Describe the following events:

the sum is greater than $8$

Answer:

Sample space when a die is rolled:

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

Let E = Event of rolling a pair of dice (= Event that a die is rolled twice!) [6x6 = 36 possible outcomes]

E = [ {(x,y): x,y $\pi{100} \in$ S } ] = {(1,1), (1,2)...(1,6),(2,1).....(6,5),(6,6)}

Now,

A : the sum is greater than 8

Possible sum greater than 8 are 9, 10, 11 and 12

A = [ {(a,b): (a,b) $\pi{100} \in$ E, a+b>8 } ]= {(3,6), (4,5), (5, 4), (6,3), (4,6), (5,5), (6,4), (5,6), (6,5), (6,6)}

Question:3(b) An experiment involves rolling a pair of dice and recording the numbers that come up. Describe the following events:

$2$ occurs on either die

Answer:

Sample space when a die is rolled:

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

Let E = Event of rolling a pair of dice (= Event that a die is rolled twice!) [6x6 = 36 possible outcomes]

E = [ {(x,y): x,y $\pi{100} \in$ S } ] = {(1,1), (1,2)...(1,6),(2,1).....(6,5),(6,6)}

Now,

B: 2 occurs on either die

Hence the number 2 can come on first die or second die or on both the die simultaneously.

B = [ {(a,b): (a,b) $\pi{100} \in$ E, a or b = 2 } ]= {(1,2), (2,2), (3, 2), (4,2), (5,2), (6,2), (2,1), (2,3), (2,4), (2,5), (2,6)}

Question:3(c). An experiment involves rolling a pair of dice and recording the numbers that come up. Describe the following events:

the sum is at least $7$ and a multiple of $3$

Answer:

Sample space when a die is rolled:

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

Let E = Event of rolling a pair of dice (= Event that a die is rolled twice!) [6x6 = 36 possible outcomes]

E = [ {(x,y): x,y $\pi{100} \in$ S } ] = {(1,1), (1,2)...(1,6),(2,1).....(6,5),(6,6)}

Now,

C: the sum is at least 7 and a multiple of 3

The sum can only be 9 or 12.

C = [ {(a,b): (a,b) $\pi{100} \in$ E, a+b>6 & a+b = 3k, k $\pi{100} \in$ I} ]= {(3,6), (6,3), (5, 4), (4,5), (6,6)}

Question:3(d). An experiment involves rolling a pair of dice and recording the numbers that come up. Describe the following events:

Which pairs of these events are mutually exclusive?

Answer:

For two elements to be mutually exclusive, there should not be any common element amongst them.

Also, A = {(3,6), (4,5), (5, 4), (6,3), (4,6), (5,5), (6,4), (5,6), (6,5), (6,6)}

B = {(1,2), (2,2), (3, 2), (4,2), (5,2), (6,2), (2,1), (2,3), (2,4), (2,5), (2,6)}

C = {(3,6), (6,3), (5, 4), (4,5), (6,6)}

Now, A $\cap$ B = $\phi$ (no common element in A and B)

Hence, A and B are mutually exclusive

Again, B $\cap$ C = $\phi$ (no common element in B and C)

Hence, B and C are mutually exclusive

Again, C $\cap$ A = {(3,6), (6,3), (5, 4), (4,5), (6,6)}

Therefore,

A and B, B and C are mutually exclusive.

Question:4(i) 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

mutually exclusive?

Answer:

Sample space when three coins are tossed = [Sample space when a coin is tossed thrice!]

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

Now,

A = Event that three heads show up = {HHH}

B = Event that two heads and one tail show up = {HHT, HTH, THH}

C = Event that three tails show up = {TTT}

D = Event that a head shows on the first coin = {HHH, HHT, HTH, HTT}

(i). For two elements X and Y to be mutually exclusive, X $\cap$ Y = $\phi$

A $\cap$ B = {HHH} $\cap$ {HHT, HTH, THH} = $\phi$ ; Hence A and B are mutually exclusive.

B $\cap$ C = {HHT, HTH, THH} $\cap$ {TTT} = $\phi$ ; Hence B and C are mutually exclusive.

C $\cap$ D = {TTT} $\cap$ {HHH, HHT, HTH, HTT} = $\phi$ ; Hence C and D are mutually exclusive.

D $\cap$ A = {HHH, HHT, HTH, HTT} $\cap$ {HHH} = {HHH} ; Hence D and A are not mutually exclusive.

A $\cap$ C = {HHH} $\cap$ {TTT} = $\phi$ ; Hence A and C are mutually exclusive.

B $\cap$ D = {HHT, HTH, THH} $\cap$ {HHH, HHT, HTH, HTT} = {HHT, HTH} ; Hence B and D are not mutually exclusive.

Question:4.(ii) 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

simple?

Answer:

Sample space when three coins are tossed = [Sample space when a coin is tossed thrice!]

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

Now,

A = Event that three heads show up = {HHH}

B = Event that two heads and one tail show up = {HHT, HTH, THH}

C = Event that three tails show up = {TTT}

D = Event that a head shows on the first coin = {HHH, HHT, HTH, HTT}

(ii).If an event X has only one sample point of a sample space, it is called a simple event.

A = {HHH} and C = {TTT}

Hence, A and C are simple events.

Question:4.(iii) 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

Compound?

Answer:

Sample space when three coins are tossed = [Sample space when a coin is tossed thrice!]

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

Now,

A = Event that three heads show up = {HHH}

B = Event that two heads and one tail show up = {HHT, HTH, THH}

C = Event that three tails show up = {TTT}

D = Event that a head shows on the first coin = {HHH, HHT, HTH, HTT}

(iv). If an event has more than one sample point, it is called a Compound event.

B = {HHT, HTH, THH} and D = {HHH, HHT, HTH, HTT}

Hence, B and D are compound events.

Question:5(i) Three coins are tossed. Describe

Two events which are mutually exclusive.

Answer:

Sample space when three coins are tossed = [Sample space when a coin is tossed thrice!]

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

(i)

A = Event that three heads show up = {HHH}

B = Event that three tails show up = {TTT}

A $\cap$ B = {HHH} $\cap$ {TTT} = $\phi$ ; Hence A and B are mutually exclusive.

Question:5(ii) Three coins are tossed. Describe

Three events which are mutually exclusive and exhaustive.

Answer:

Sample space when three coins are tossed = [Sample space when a coin is tossed thrice!]

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

Let ,

A = Getting no tails = {HHH}

B = Getting exactly one tail = {HHT, HTH, THH}

C = Getting at least two tails = {HTT, THT, TTH}

Clearly, A $\cap$ B = $\phi$ ; B $\cap$ C = $\phi$ ; C $\cap$ A = $\phi$

Since (A and B), (B and C) and (A and C) are mutually exclusive

Therefore A, B and C are mutually exclusive.

Also,

A $\cup$ B $\cup$ C = S

Hence A, B and C are exhaustive events.

Hence, A, B and C are three events which are mutually exclusive and exhaustive.

Question:5(iii). Three coins are tossed. Describe

Two events, which are not mutually exclusive.

Answer:

Sample space when three coins are tossed = [Sample space when a coin is tossed thrice!]

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

Let ,

A = Getting at least one head = {HHH, HHT, HTH, THH, TTH}

B = Getting at most one head = {TTH, TTT}

Clearly, A $\cap$ B = {TTH} $\neq$ $\phi$

Hence, A and B are two events which are not mutually exclusive.

Question:5.(iv) Three coins are tossed. Describe

Two events which are mutually exclusive but not exhaustive.

Answer:

Sample space when three coins are tossed = [Sample space when a coin is tossed thrice!]

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

Let ,

A = Getting exactly one head = {HTT, THT, TTH}

B = Getting exactly one tail = {HHT, HTH, THH}

Clearly, A $\cap$ B = $\phi$

Hence, A and B are mutually exclusive.

Also, A $\cup$ B $\neq$ S

Hence, A and B are not exhaustive.

Question:5.(v) Three coins are tossed. Describe

Three events which are mutually exclusive but not exhaustive

Answer:

Sample space when three coins are tossed = [Sample space when a coin is tossed thrice!]

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

Let ,

A = Getting exactly one tail = {HHT, HTH, THH}

B = Getting exactly two tails = {HTT, TTH, THT}

C = Getting exactly three tails = {TTT}

Clearly, A $\cap$ B = $\phi$ ; B $\cap$ C = $\phi$ ; C $\cap$ A = $\phi$

Since (A and B), (B and C) and (A and C) are mutually exclusive

Therefore A, B and C are mutually exclusive.

Also,

A $\cup$ B $\cup$ C = {HHT, HTH, THH, HTT, TTH, THT, TTT} $\neq$ S

Hence A, B and C are not exhaustive events.

Question:6.(i) 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 $\leq 5$.

Describe the events

$A{}'$

Answer:

Sample space when two dice are thrown:

S = {(x,y): 1 $\leq$ x,y $\leq$ 6}

A: getting an even number on the first die = {(a,b): a $\in$ {2,4,6} and 1 $\leq$ b $\leq$ 6}

= {(2, 1), (2,2), (2,3), (2,4), (2,5), (2,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}

(i) Therefore, A'= {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6)}

= B : getting an odd number on the first die.

Question:6.(ii) 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 $\leq 5$.

Describe the events

not B

Answer:

Sample space when two dice are thrown:

S = {(x,y): 1 $\leq$ x,y $\leq$ 6}

B: getting an odd number on the first die = {(a,b): a $\in$ {1,3,5} and 1 $\leq$ b $\leq$ 6}

= {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6)}

(ii) Therefore, B'= {(2, 1), (2,2), (2,3), (2,4), (2,5), (2,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}

= A : getting an even number on the first die.

Question:6.(iii) 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 $\leq 5$.

Describe the events

A or B

Answer:

Sample space when two dice are thrown:

S = {(x,y): 1 $\leq$ x,y $\leq$ 6}

A: getting an even number on the first die = {(a,b): a $\in$ {2,4,6} and 1 $\leq$ b $\leq$ 6}

= {(2, 1), (2,2), (2,3), (2,4), (2,5), (2,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}

B: getting an odd number on the first die = {(a,b): a $\in$ {1,3,5} and 1 $\leq$ b $\leq$ 6}

= {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6)}

(iii) A or B = A $\cup$ B = {(1,1), (1,2) .... (1,6), (3,1), (3,2).... (3,6), (5,1), (5,2)..... (5,6), (2,1), (2,2)..... (2,6), (4,1), (4,2)..... (4,6), (6,1), (6,2)..... (6,6)} = S

Question:6.(iv) 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 $\leq 5$

Describe the events

A and B

Answer:

Sample space when two dice are thrown:

S = {(x,y): 1 $\leq$ x,y $\leq$ 6}

A: getting an even number on the first die = {(a,b): a $\in$ {2,4,6} and 1 $\leq$ b $\leq$ 6}

= {(2, 1), (2,2), (2,3), (2,4), (2,5), (2,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}

B: getting an odd number on the first die = {(a,b): a $\in$ {1,3,5} and 1 $\leq$ b $\leq$ 6}

= {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6)}

(iii) A and B = A $\cap$ B = A $\cap$ A' = $\phi$ (From (ii))

Question:6.(v) 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 $\leq 5$

Describe the events

A but not C

Answer:

Sample space when two dice are thrown:

S = {(x,y): 1 $\leq$ x,y $\leq$ 6}

A: getting an even number on the first die = {(a,b): a $\in$ {2,4,6} and 1 $\leq$ b $\leq$ 6}

= {(2, 1), (2,2), (2,3), (2,4), (2,5), (2,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}

C: getting the sum of the numbers on the dice $\leq$ 5

The possible sum are 2,3,4,5

C = {(a,b): 2 $\leq$ a + b $\leq$ 5} = {(1, 1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (3,1), (3,2), (4,1)}

(v) A but not C = A - C = {(2,4), (2,5), (2,6), (4,2), (4,3), (4,4), (4,5), (4,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}

Question:6.(vi) 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 $\leq 5$

Describe the events

B or C

Answer:

Sample space when two dice are thrown:

S = {(x,y): 1 $\leq$ x,y $\leq$ 6}

B: getting an odd number on the first die = {(a,b): a $\in$ {1,3,5} and 1 $\leq$ b $\leq$ 6}

= {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6)}

C: getting the sum of the numbers on the dice $\leq$ 5

The possible sum are 2,3,4,5

C = {(a,b): 2 $\leq$ a + b $\leq$ 5} = {(1, 1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (3,1), (3,2), (4,1)}

(vi) B or C = B $\cup$ C = {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6)}

Question:6.(vii) 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 $\leq 5$

Describe the events

B and C

Answer:

Sample space when two dice are thrown:

S = {(x,y): 1 $\leq$ x,y $\leq$ 6}

B: getting an odd number on the first die = {(a,b): a $\in$ {1,3,5} and 1 $\leq$ b $\leq$ 6}

= {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6)}

C: getting the sum of the numbers on the dice $\leq$ 5

The possible sum are 2,3,4,5

C = {(a,b): 2 $\leq$ a + b $\leq$ 5} = {(1, 1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (3,1), (3,2), (4,1)}

(vi) B and C = B $\cap$ C = {(1, 1), (1,2), (1,3), (1,4), (3,1), (3,2)}

Question:6.(viii) 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 $\leq 5$

Describe the events

$A\cap {B}'\cap {C}'$

Answer:

Sample space when two dice are thrown:

S = {(x,y): 1 $\leq$ x,y $\leq$ 6}

A: getting an even number on the first die = {(a,b): a $\in$ {2,4,6} and 1 $\leq$ b $\leq$ 6}

= {(2, 1), (2,2), (2,3), (2,4), (2,5), (2,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}

B: getting an odd number on the first die = {(a,b): a $\in$ {1,3,5} and 1 $\leq$ b $\leq$ 6}

= {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6)}

C: getting the sum of the numbers on the dice $\leq$ 5

The possible sum are 2,3,4,5

C = {(a,b): 2 $\leq$ a + b $\leq$ 5} = {(1, 1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (3,1), (3,2), (4,1)}

(viii) A $\cap$ B' $\cap$ C' = A $\cap$ A $\cap$ C' (from (ii))

= A $\cap$ C' = A - C = {(2,4), (2,5), (2,6), (4,2), (4,3), (4,4), (4,5), (4,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}

Question:7.(i) Refer to question 6 above, state true or false: (give reason for your answer)

A and B are mutually exclusive

Answer:

Here,

A = {(2, 1), (2,2), (2,3), (2,4), (2,5), (2,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}

B = {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6)}

(i) X and Y are mutually exclusive if and only if X $\cap$ Y = $\phi$

A $\cap$ B = $\phi$ , since A and B have no common element amongst them.

Hence, A and B are mutually exclusive. TRUE

Question:7.(ii) Refer to question 6 above, state true or false: (give reason for your answer)

A and B are mutually exclusive and exhaustive

Answer:

Here,

A = {(2, 1), (2,2), (2,3), (2,4), (2,5), (2,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}

B = {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6)}

(ii) X and Y are mutually exclusive if and only if X $\cap$ Y = $\phi$

A $\cap$ B = $\phi$ , since A and B have no common element amongst them.

Hence, A and B are mutually exclusive.

Also,

A $\cup$ B = {(2,1), (2,2).... (2,6), (4,1), (4,2).....(4,6), (6,1), (6,2)..... (6,6), (1,1), (1,2).... (1,6), (3,1), (3,2)..... (3,6), (5,1), (5,2).... (5,6)} = S

Hence, A and B are exhaustive.

TRUE

Question:7.(iii) Refer to question 6 above, state true or false: (give reason for your answer)

$A=B{}'$

Answer:

Here,

S = {(x,y): 1 $\leq$ x,y $\leq$ 6}

A = {(2, 1), (2,2), (2,3), (2,4), (2,5), (2,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}

B = {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6)}

(iii) Therefore, B' = S -B = {(2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)} = A

TRUE

Question:7.(iv) Refer to question 6 above, state true or false: (give reason for your answer)

A and C are mutually exclusive

Answer:

Here,

S = {(x,y): 1 $\leq$ x,y $\leq$ 6}

A = {(2, 1), (2,2), (2,3), (2,4), (2,5), (2,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}

C = {(1, 1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (3,1), (3,2), (4,1)}

(iv) X and Y are mutually exclusive if and only if X $\cap$ Y = $\phi$

A $\cap$ C = {(2,1), (2,2), (2,3), (4,1)} ,

Hence, A and B are not mutually exclusive. FALSE

Question:7.(v) Refer to question 6 above, state true or false: (give reason for your answer)

$A$ and ${B}'$ are mutually exclusive.

Answer:

X and Y are mutually exclusive if and only if X $\cap$ Y = $\phi$

A $\cap$ B' = A $\cap$ A = A (From (iii))

$\therefore$ A $\cap$ B’ $\neq \phi$

Hence A and B' not mutually exclusive. FALSE

Question:7.(vi) Refer to question 6 above, state true or false: (give reason for your answer)

${A}',{B}',C$ are mutually exclusive and exhaustive.

Answer:

Here,

S = {(x,y): 1 $\leq$ x,y $\leq$ 6}

A = {(2, 1), (2,2), (2,3), (2,4), (2,5), (2,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}

B = {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6)}

C = {(1, 1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (3,1), (3,2), (4,1)}

(vi) X and Y are mutually exclusive if and only if X $\cap$ Y = $\phi$

$\therefore$ A' $\cap$ B' = B $\cap$ A = $\phi$ (from (iii) and (i))

Hence A' and B' are mutually exclusive.

Again,

$\therefore$ B' $\cap$ C = A $\cap$ C $\neq$ $\phi$ (from (iv))

Hence B' and C are not mutually exclusive.

Hence, A', B' and C are not mutually exclusive and exhaustive. FALSE

Also read

Topics covered in Chapter 14 Probability Exercise 14.1

1) What is an Event in Probability?

An event in probability is the set of outcomes of a random experiment. It is any subset of the sample space, including one outcome, multiple outcomes, or even no outcome.

2) The Occurrence of an Event

An event occurs if the outcome of the experiment is one of the outcomes in that event. For example, getting a 4 when rolling a die is the occurrence of the event "rolling an even number."

3) Types of Events

Events can be classified as:

  • Simple or elementary
  • Compound
  • Impossible
  • Sure
  • Mutually exclusive
  • Exhaustive
  • Complementary

4) Impossible and Sure Events

  • Impossible Event: An event that can never occur. Example: Getting a 7 on a standard six-sided die. Probability = 0.
  • Sure Event: An event that always occurs. Example: Getting a number less than 7 on a six-sided die. Probability = 1.

5) Simple Event

A simple event consists of only one outcome. Example: Getting a 3 when a die is rolled.

6) Compound Event

A compound event includes two or more simple events. Example: Getting an even number (2, 4, or 6) when a die is rolled.

7)Algebra of Events

The algebra of events includes operations like:

  • Union (A ∪ B): Event A or B occurs.
  • Intersection (A ∩ B): Both A and B occur.
  • Complement (A'): A does not occur.
  • Difference (A – B): A occurs but B does not.

8) Complementary Event

The complement of event A (written as A') includes all outcomes not in A. Together, A and A' make up the entire sample space.

9) The Event ‘A or B’

This is the union of events A and B (A ∪ B). It includes all outcomes that are in A, B, or both.

10) The Event ‘A but not B’

This is the difference between A and B (A – B). It includes outcomes that are in A but not in B.

11) Mutually Exclusive Events

Events are mutually exclusive if they cannot happen at the same time. Example: Getting a 2 and a 5 in one roll of a die.

12) Exhaustive Events

Events are exhaustive if their union includes the entire sample space. That means one of them must occur. Example: In a die roll, the events {1}, {2}, {3}, {4}, {5}, {6} are exhaustive.

Also Read

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Frequently Asked Questions (FAQs)

Q: A die is rolled then what is the probability of getting number 6 on the die face ?
A:

The probability of getting number 6 on the die face is = 1/6

Q: A die is rolled then what is the probability of getting number 7 on the die face ?
A:

The probability of getting the number '7' on the die face is zero.

Q: When a biased coin is tossed and the probability of getting head on the coin is 0.61, then what is the probability of getting tail ?
A:

The probability of getting head = 0.61

Probability of getting tail = 1- 0.61 = 0.39

Q: What is the probability of a sure event ?
A:

The probability of a sure event is 1.

Q: what is the probability of an impossible event ?
A:

The probability of an impossible event is zero.

Q: If the probability of an event A is 0.64 then what is the probability of its compliment event ?
A:

The probability of compliment of A p(A') = 1-p(A) = 0.36.

Q: If the probability of an event A is zero the does A is an impossible event ?
A:

Yes, If the probability of an event A is zero then A is an impossible event.

Q: What is equal likely outcomes ?
A:

If the probability of all outcomes of a random experiment is the same then such outcomes are called equal likely outcomes.

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