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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$.
Describe the events (i) A′
Solution :
Step 1: Defining the Sample Space ($S$)
When two six-faced dice are rolled, the sample space $S$ consists of all possible ordered pairs $(x, y)$, where $x$ represents the result of the first die and $y$ represents the result of the second die. Since each die has 6 faces, the total number of outcomes is $6 \times 6 = 36$.
$S = \{(x, y) : x, y \in \{1, 2, 3, 4, 5, 6\}\}$
Step 2: Defining Event $A$
Event $A$ is defined as "getting an even number on the first die." The possible even numbers for the first die are $\{2, 4, 6\}$.
$A = \{(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6),$
$(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6),$
$(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)\}$
[By Definition of the Complementary Event: The complement $A'$ consists of all outcomes in $S$ that are not in $A$.]
Step 3: Calculating the Complement Event $A'$
If $A$ is the set of outcomes where the first die is even $\{2, 4, 6\}$, then the complement $A'$ is the set of outcomes where the first die is not even. Given the sample space, if the first die is not even, it must be odd $\{1, 3, 5\}$.
$A' = \{(x, y) \in S : x \in \{1, 3, 5\}\}$
$A' = \{(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6),$
$(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6),$
$(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6)\}$
Visual Representation of Event Space
Step 4: Logical Conclusion
The complement event $A'$ represents the occurrence of an odd number on the first die. This is mathematically equivalent to the event $B$ provided in the problem statement ($B$: getting an odd number on the first die). Thus, $A' = B$.
Final Solution: $A' = \{(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6)\}$
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(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
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