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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$.
Describe the events (iv) A and B
Solution :
Step 1: Defining the Sample Space ($S$)
When two six-sided dice are thrown, each die has 6 outcomes ($1, 2, 3, 4, 5, 6$). The total number of outcomes in the sample space $S$ is $6 \times 6 = 36$.
$S = \{ (x, y) : x, y \in \{1, 2, 3, 4, 5, 6\} \}$
Step 2: Defining Event A and Event B
- Event A: Getting an even number on the first die. $A = \{ (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6) \}$
- Event B: Getting an odd number on the first die. $B = \{ (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6) \}$
Step 3: Analyzing "A and B" (Intersection $A \cap B$)
The event "A and B" denotes the intersection of set $A$ and set $B$ ($A \cap B$). This event consists of outcomes that belong to both Event A and Event B simultaneously [Per the Definition of Intersection in Set Theory].
Step 4: Logical Evaluation of the Intersection
An outcome $(x, y)$ belongs to $A \cap B$ if $x$ is simultaneously even and odd.
Since the set of even numbers $\{2, 4, 6\}$ and the set of odd numbers $\{1, 3, 5\}$ are mutually exclusive, there exists no element $x$ such that $x \in \{2, 4, 6\} \cap \{1, 3, 5\}$.
Therefore, $A \cap B = \emptyset$ [By the Property of Mutually Exclusive Events].
Step 5: Conclusion on Event (A and B)
Since the intersection results in an empty set, the event "A and B" is an impossible event. There are no outcomes in the sample space where the first die is both even and odd.
Final Solution: $A \cap B = \phi$ (The event is impossible).
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(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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