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Q5(v):
Three coins are tossed. Describe (v) Three events which are mutually exclusive but not exhaustive.
Solution :
Step 1: Establishing the Sample Space ($S$)
When three coins are tossed, each coin has two possible outcomes: Head ($H$) or Tail ($T$). For $n=3$ coins, the total number of elementary events in the sample space $S$ is given by $2^n = 2^3 = 8$.
$S = \{HHH, HHT, HTH, THH, HTT, THT, TTH, TTT\}$
Step 2: Defining Mutually Exclusive and Exhaustive Conditions
- Mutually Exclusive: Events $A, B,$ and $C$ are mutually exclusive if $A \cap B = \phi$, $A \cap C = \phi$, and $B \cap C = \phi$. That is, no two events can occur simultaneously.
- Exhaustive: Events are exhaustive if their union is equal to the sample space: $A \cup B \cup C = S$.
- Requirement: We require events that are mutually exclusive but not exhaustive, meaning $A \cup B \cup C \subset S$.
Step 3: Constructing the Events
Let us define three specific events based on the number of heads appearing in the toss:
- Event $E_1$: Getting exactly 3 heads.
$E_1 = \{HHH\}$ - Event $E_2$: Getting exactly 3 tails.
$E_2 = \{TTT\}$ - Event $E_3$: Getting exactly 2 heads and 1 tail.
$E_3 = \{HHT, HTH, THH\}$
Step 4: Verification of Mutually Exclusive Property
We check for intersections between the defined events:
| Intersection | Result | Status |
|---|---|---|
| $E_1 \cap E_2$ | $\phi$ | Mutually Exclusive |
| $E_1 \cap E_3$ | $\phi$ | Mutually Exclusive |
| $E_2 \cap E_3$ | $\phi$ | Mutually Exclusive |
[By Definition of Disjoint Sets: Sets are mutually exclusive if they share no common elements].
Step 5: Verification of Non-Exhaustive Property
Calculate the union of the three events:
$E_1 \cup E_2 \cup E_3 = \{HHH, TTT, HHT, HTH, THH\}$
Comparing this to the Sample Space $S$:
$\{HHH, TTT, HHT, HTH, THH\} \neq \{HHH, HHT, HTH, THH, HTT, THT, TTH, TTT\}$
The elements $\{HTT, THT, TTH\}$ are missing from the union. Since $E_1 \cup E_2 \cup E_3 \subset S$, the events are not exhaustive.
Visual Representation of Sample Space Distribution
Final Solution: The three events $E_1 = \{HHH\}$, $E_2 = \{TTT\}$, and $E_3 = \{HHT, HTH, THH\}$ are mutually exclusive because they share no common elements, and they are not exhaustive because their union does not contain the elements $\{HTT, THT, TTH\}$.
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.
- 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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