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Q2(iii):
A die is thrown. Describe the following events: (iii) C: a multiple of 3
Solution :
Step 1: Identification of the Sample Space
When a standard fair six-sided die is thrown, the sample space $S$ is the set of all possible outcomes. Since the faces of the die are numbered 1 through 6, we define $S$ as:
$S = \{1, 2, 3, 4, 5, 6\}$
[Per the classical definition of probability, the sample space is the exhaustive set of all elementary outcomes of a random experiment.]
Step 2: Defining the Condition for Event C
The event $C$ is defined as the occurrence of a "multiple of 3" upon the throw of the die. To identify the elements of $C$, we analyze each element $x \in S$ to determine if $x$ is divisible by 3 [By the definition of divisibility, an integer $a$ is a multiple of $b$ if $a = bk$ for some integer $k$].
- For $x = 1$: $1 \div 3$ is not an integer.
- For $x = 2$: $2 \div 3$ is not an integer.
- For $x = 3$: $3 = 3 \times 1$; therefore, 3 is a multiple of 3.
- For $x = 4$: $4 \div 3$ is not an integer.
- For $x = 5$: $5 \div 3$ is not an integer.
- For $x = 6$: $6 = 3 \times 2$; therefore, 6 is a multiple of 3.
Step 3: Construction of Event Set C
Collecting the elements identified in the previous step, we form the set $C$ as a subset of the sample space $S$:
$C = \{3, 6\}$
Step 4: Visual Representation of the Event
Final Solution: The event C, representing a multiple of 3, is the set $C = \{3, 6\}$.
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(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(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) $A = B'$
- 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
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