Find the best tutors and institutes for Class 11 Tuition
Q7(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 ≤ 5.
state true or false: (give reason for your answer) (iii) $A = B'$
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.
state true or false: (give reason for your answer) (iii) $A = B'$
Solution :
Step 1: Defining the Sample Space ($S$)
When two dice are thrown simultaneously, each die has 6 faces $\{1, 2, 3, 4, 5, 6\}$. The sample space $S$ consists of all ordered pairs $(x, y)$, where $x$ is the outcome of the first die and $y$ is the outcome of the second die. The total number of outcomes is $n(S) = 6 \times 6 = 36$.
Step 2: Defining Event A and Event B
Event $A$ is defined as "getting an even number on the first die." The possible outcomes for $x$ are $\{2, 4, 6\}$. The outcomes for $y$ can be any integer from $1$ to $6$. $A = \{(2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)\}$. [Cardinality: $n(A) = 3 \times 6 = 18$].
Event $B$ is defined as "getting an odd number on the first die." The possible outcomes for $x$ are $\{1, 3, 5\}$. The outcomes for $y$ can be any integer from $1$ to $6$. $B = \{(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6)\}$. [Cardinality: $n(B) = 3 \times 6 = 18$].
Step 3: Determining the Complement of Event B ($B'$)
The complement event $B'$ consists of all outcomes in the sample space $S$ that are not in event $B$. By definition, $B' = S - B$. Since $S$ is the set of all pairs $(x, y)$ where $x \in \{1, 2, 3, 4, 5, 6\}$, and $B$ contains all pairs where $x \in \{1, 3, 5\}$, then $B'$ must contain all pairs where $x \in \{2, 4, 6\}$.
Step 4: Logical Comparison
As derived in Step 3, the outcomes belonging to $B'$ are exactly the pairs where the first die shows an even number ($x \in \{2, 4, 6\}$). Comparing the set $A$ (defined in Step 2) and the set $B'$ (derived in Step 3): $A = \{(x, y) \mid x \in \{2, 4, 6\}, y \in \{1, \dots, 6\}\}$ $B' = \{(x, y) \mid x \in \{2, 4, 6\}, y \in \{1, \dots, 6\}\}$
Since $A = B'$, the statement is logically valid. [By the Law of Complementation in Set Theory, since the sample space is partitioned into $A$ (even first) and $B$ (odd first), $B^c$ is the set of all outcomes not in $B$, which is $A$.]
Final Solution: True. The event $A$ (getting an even number on the first die) is identical to the complement of event $B$ (getting an odd number on the first die) because the first die can only result in either an even or an odd number, and these events are mutually exclusive and collectively exhaustive.
More Questions from Class 11 Mathematics Probability EXERCISE 14.1
- Q1: 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?
- Q2(i): A die is thrown. Describe the following events: (i) A: a number less than 7
- Q2(ii): A die is thrown. Describe the following events: (ii) B: a number greater than 7
- Q2(iii): A die is thrown. Describe the following events: (iii) C: a multiple of 3
- Q2(iv): A die is thrown. Describe the following events: (iv) D: a number less than 4
- Q2(v): A die is thrown. Describe the following events: (v) E: an even number greater than 4
- Q2(vi): A die is thrown. Describe the following events: (vi) F: a number not less than 3. Also find $A \cup B$, $A \cap B$, $B \cup C$, $E \cap F$, $D \cap E$, $A – C$, $D – E$, $E \cap F'$, $F'$
- Q3: 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?
- Q4(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 (i) mutually exclusive?
- Q4(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 (ii) simple?
- Q4(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 (iii) Compound?
- Q5(i): Three coins are tossed. Describe (i) Two events which are mutually exclusive.
- Q5(ii): Three coins are tossed. Describe (ii) Three events which are mutually exclusive and exhaustive.
- Q5(iii): Three coins are tossed. Describe (iii) Two events, which are not mutually exclusive.
- Q5(iv): Three coins are tossed. Describe (iv) Two events which are mutually exclusive but not exhaustive.
- Q5(v): Three coins are tossed. Describe (v) Three events which are mutually exclusive but not exhaustive.
- Q6(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 $ \le 5$. Describe the events (i) A′
- Q6(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 $ \le 5$. Describe the events (ii) not B
- Q6(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 $ \le 5$. Describe the events (iii) A or B
- Q6(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 $ \le 5$. Describe the events (iv) A and B
- Q6(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 $ \le 5$. Describe the events (v) A but not C
- Q6(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 $ \le 5$. Describe the events (vi) B or C
- Q6(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 $ \le 5$. Describe the events (vii) B and C
- Q6(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 $ \le 5$. Describe the events (viii) $A \cap B' \cap C'$
- Q7(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 ≤ 5. state true or false: (give reason for your answer) (i) A and B are mutually exclusive
- Q7(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 ≤ 5. state true or false: (give reason for your answer) (ii) A and B are mutually exclusive and exhaustive
- Q7(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 ≤ 5. state true or false: (give reason for your answer) (iv) A and C are mutually exclusive
- Q7(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 ≤ 5. state true or false: (give reason for your answer) (v) $A$ and $B'$ are mutually exclusive.
- Q7(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 ≤ 5. state true or false: (give reason for your answer) (vi) $A'$, $B'$, $C$ are mutually exclusive and exhaustive.
CBSE Solutions for Class 11 Mathematics Probability
Chapters in CBSE - Class 11 Mathematics
Top Tutors who teach Probability
Top Tutor
Gadarpur, Gadarpur
5 yrs of
Exp
2 students
144 hrs of classes completed
500
per hour
Online Classes
I am a teacher with 7 years of experience in teaching Accountancy at school (ISC & CBSE). I have completed my graduation (Bcom) from Kurukshetra University and Post Graduation (Mcom) from IGNOU with a diploma in International Business Operations.
Nidhi Ma'am explained the topic till I understood it. She clarified all my doubts and made the topic very clear. I thoroughly enjoyed having classes with her.
Top Tutor
Vile Parle East, Mumbai
10 yrs of
Exp
5 students
201 hrs of classes completed
500
per hour
Online Classes
Tutor's home
Chemistry class 11 and cbse and state board free lance tutor online as well as teaching at my home
She was very good in explaining my 11th grade basics are strong because of her. She was very supportive and understanding.
Top Tutor
Bajaj Nagar, Nagpur
3 yrs of
Exp
Online Classes
Student's Home
Only the students attitude makes a subject HARD or EASY. FUNDAMENTALS are the only need for all Competitive Exams. Once you GET it, cant be bothered again.
Top Tutor
Talwandi Sector-B, Kota
10 yrs of
Exp
13 students
891 hrs of classes completed
1850
per hour
Online Classes
Student's Home
Tutor's home
My teaching exp.11 year in most reputed coaching in india. Mentor of jee top rank air-12 yatis agrawal, 19, 28 many more in top 100 in jee adavnced. Many of students got 100%tile in jee mains along with 100% marks in board exams. I also provide complete relevent study material like worksheet, practice sheet , dpps , assignment etc. Daily quizz based on assignment. Periodically test papers topic wise, minor and major as per shedule. I have helped thousands of students overcome their fear of maths.
He is a very nice sir and knows his knowledge well, he is also well educated and good with his content.
Top Tutor
Shastri Colony, Panipat
7 yrs of
Exp
17 students
898 hrs of classes completed
500
per hour
Online Classes
Tutor's home
Experienced Educator specializing in Physics. I have entered into the field of teaching and started to make and feel physics easy and exciting by the students and also make students to crack the most of the competitive examinations by which I will make them achieve their goals in future. I have been successfully training students from class 10t,11th and 12th for the last 11 years in Physics, along with JEE Mains & Advanced, NEET exam preparation. With 11 years of teaching experience and a background in Unacademy ( Doubt Solving Expert) and Filo Edutech Platform, I provide comprehensive coaching for students preparing for board exams and competitive tests.
A good, kind and a patient teacher, who creates a positive and a supportive environment for her students.
