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Q2(vi):
A die is thrown. Describe the following events: (vi) F: a number not less than 3. Also find $A \cup B$, $A \cap B$, $B \cup C$, $E \cap F$, $D \cap E$, $A – C$, $D – E$, $E \cap F'$, $F'$
Solution :
Step 1: Defining the Sample Space
For a single throw of a fair, six-faced die, the sample space $S$ consists of all possible outcomes. Each face is numbered from 1 to 6.
$S = \{1, 2, 3, 4, 5, 6\}$
Step 2: Defining the Given Events
To compute the requested set operations, we must first define the standard events for a die throw (based on typical textbook conventions for this problem set):
- $A = \{x : x < 1\} = \emptyset$ (Impossible event)
- $B = \{x : 1 \leq x \leq 3\} = \{1, 2, 3\}$
- $C = \{x : x > 3\} = \{4, 5, 6\}$
- $D = \{x : x \geq 5\} = \{5, 6\}$
- $E = \{x : x \text{ is even}\} = \{2, 4, 6\}$
- $F = \{x : x \geq 3\} = \{3, 4, 5, 6\}$ [Given by question (vi)]
Step 3: Executing Set Operations
| Operation | Calculation | Result |
|---|---|---|
| $A \cup B$ | $\emptyset \cup \{1, 2, 3\}$ [Identity Law of Union] | $\{1, 2, 3\}$ |
| $A \cap B$ | $\emptyset \cap \{1, 2, 3\}$ [Null set property] | $\emptyset$ |
| $B \cup C$ | $\{1, 2, 3\} \cup \{4, 5, 6\}$ | $\{1, 2, 3, 4, 5, 6\}$ |
| $E \cap F$ | $\{2, 4, 6\} \cap \{3, 4, 5, 6\}$ | $\{4, 6\}$ |
| $D \cap E$ | $\{5, 6\} \cap \{2, 4, 6\}$ | $\{6\}$ |
| $A - C$ | $\emptyset \setminus \{4, 5, 6\}$ [Difference definition] | $\emptyset$ |
| $D - E$ | $\{5, 6\} \setminus \{2, 4, 6\}$ | $\{5\}$ |
Step 4: Complementary and Conditional Intersections
Finding $F'$: The complement of $F$ is the set of elements in $S$ not in $F$.
$F' = S \setminus F = \{1, 2, 3, 4, 5, 6\} \setminus \{3, 4, 5, 6\} = \{1, 2\}$
Finding $E \cap F'$:
$E \cap F' = \{2, 4, 6\} \cap \{1, 2\} = \{2\}$
Step 5: Visual Representation of Event F
Final Solution: The outcomes for $F$ are $\{3, 4, 5, 6\}$. The calculated set results are: $A \cup B = \{1, 2, 3\}$, $A \cap B = \emptyset$, $B \cup C = \{1, 2, 3, 4, 5, 6\}$, $E \cap F = \{4, 6\}$, $D \cap E = \{6\}$, $A - C = \emptyset$, $D - E = \{5\}$, $E \cap F' = \{2\}$, and $F' = \{1, 2\}$.
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
- 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
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