GSEB Class 11 Maths Solutions Chapter 16 Probability Exercise 16.2

Question 1. 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?
Answer: The possible outcomes when we roll a die are given by the sample space \( S = \{1, 2, 3, 4, 5, 6\} \).
Event E, where the die shows a 4, is \( E = \{4\} \).
Event F, where the die shows an even number, is \( F = \{2, 4, 6\} \).
The intersection of events E and F is \( E \cap F = \{4\} \).
\( \implies \) Since this intersection \( E \cap F \) is not an empty set \( (\emptyset) \), events E and F share a common outcome.
\( \implies \) Therefore, E and F are not mutually exclusive events.
In simple words: Two events are mutually exclusive if they cannot happen at the same time. Here, the number 4 is in both event E (showing 4) and event F (showing an even number). Since they share an outcome, they are not mutually exclusive.

Exam Tip: To check if events are mutually exclusive, always find their intersection. If the intersection is empty (\( \emptyset \)), the events are mutually exclusive; otherwise, they are not.

Question 2. A die is thrown. Describe the following events:
(i) A: a number less than 7
(ii) B: a number greater than 7
(iii) C: a multiple of 3
(iv) D: a number less than 4
(v) E : an even number greater than 4
(vi) F: a number not less than 3
Also, find A U B, A ∩ B, E U F, D ∩ E, A - C, D - E, F' and E ∩ F'.
Answer: The sample space S, when a die is thrown, contains the numbers \( S = \{1, 2, 3, 4, 5, 6\} \).
(i) Event A, a number less than 7, includes all outcomes: \( A = \{1, 2, 3, 4, 5, 6\} \).
(ii) Event B, a number greater than 7, is an empty set: \( B = \{\} = \emptyset \).
(iii) Event C, a multiple of 3, is: \( C = \{3, 6\} \).
(iv) Event D, a number less than 4, is: \( D = \{1, 2, 3\} \).
(v) Event E, an even number greater than 4, is: \( E = \{6\} \).
(vi) Event F, a number not less than 3 (meaning 3 or greater), is: \( F = \{3, 4, 5, 6\} \).
Now, we will find the requested operations:
The union of A and B is: \( A \cup B = \{1, 2, 3, 4, 5, 6\} \cup \emptyset = \{1, 2, 3, 4, 5, 6\} \).
The intersection of A and B is: \( A \cap B = \{1, 2, 3, 4, 5, 6\} \cap \emptyset = \emptyset \).
The union of E and F is: \( E \cup F = \{6\} \cup \{3, 4, 5, 6\} = \{3, 4, 5, 6\} \).
The intersection of D and E is: \( D \cap E = \{1, 2, 3\} \cap \{6\} = \emptyset \).
The difference A - C (elements in A but not in C) is: \( A - C = \{1, 2, 3, 4, 5, 6\} - \{3, 6\} = \{1, 2, 4, 5\} \).
The difference D - E (elements in D but not in E) is: \( D - E = \{1, 2, 3\} - \{6\} = \{1, 2, 3\} \).
The complement of F (F') consists of elements in S but not in F: \( F' = \{1, 2, 3, 4, 5, 6\} - \{3, 4, 5, 6\} = \{1, 2\} \).
The intersection of E and F' is: \( E \cap F' = \{6\} \cap \{1, 2\} = \emptyset \).
In simple words: First, list all possible numbers when rolling a die. Then, for each event (A, B, C, D, E, F), list the numbers that fit its description. After that, use these lists to combine or compare events to find unions, intersections, and complements.

Exam Tip: Always start by clearly defining the sample space. This helps in correctly listing the elements of each event and performing set operations accurately.

Question 3. 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.
Also, find A ∩ B, B ∩ C and A ∩ C.
Are
1. A and B mutually exclusive?
2. B and C mutually exclusive?
3. A and C mutually exclusive?
Answer: When two dice are thrown, there are \( 6 \times 6 = 36 \) possible outcomes. The sample space S is:
\( S = \{(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), \)
\( (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), \)
\( (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), \)
\( (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), \)
\( (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), \)
\( (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)\} \).
Event A, where the sum is greater than 8, includes:
\( A = \{(3, 6), (4, 5), (5, 4), (6, 3), (4, 6), (5, 5), (6, 4), (5, 6), (6, 5), (6, 6)\} \).
Event B, where a 2 occurs on either die, includes:
\( B = \{(1, 2), (2, 2), (3, 2), (4, 2), (5, 2), (6, 2), (2, 1), (2, 3), (2, 4), (2, 5), (2, 6)\} \).
Event C, where the sum is at least 7 and a multiple of 3, includes:
\( C = \{(3, 6), (6, 3), (5, 4), (4, 5), (6, 6)\} \).
Now, let's find the intersections:
\( A \cap B = \emptyset \) (There are no common outcomes between A and B).
\( B \cap C = \emptyset \) (There are no common outcomes between B and C).
\( A \cap C = \{(3, 6), (6, 3), (5, 4), (4, 5), (6, 6)\} \) (These are the common outcomes for A and C).
Now, we answer whether the events are mutually exclusive:
1. Since \( A \cap B = \emptyset \), events A and B are mutually exclusive.
2. Since \( B \cap C = \emptyset \), events B and C are mutually exclusive.
3. Since \( A \cap C \neq \emptyset \), events A and C are not mutually exclusive.
In simple words: First, list all the pairs of numbers that can come up when you roll two dice. Then, for each event (A, B, C), list the pairs that fit its rule. To see if events are mutually exclusive, check if they share any pairs; if they don't, then they are mutually exclusive.

Exam Tip: For problems involving multiple dice, systematically list all possible outcomes in the sample space. This helps avoid errors when defining events and finding their intersections.

Question 4. Three coins are tossed once. Let A denotes the event "three heads show", B denotes the event "two heads and one tail shows", C denotes the event "three tails show" and D denote the event "a head shows on the first coin".
Which events are
1. mutually exclusive?
2. simple?
3. compound?
Answer: When three coins are tossed, the sample space S includes:
\( S = \{\text{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}\} \).
Event A, showing three heads, is: \( A = \{\text{HHH}\} \).
Event B, showing two heads and one tail, is: \( B = \{\text{HHT, HTH, THH}\} \).
Event C, showing three tails, is: \( C = \{\text{TTT}\} \).
Event D, showing a head on the first coin, is: \( D = \{\text{HHH, HHT, HTH, HTT}\} \).
Now, let's determine the types of events:
1. Mutually exclusive events:
To find mutually exclusive events, we check if their intersections are empty:
\( A \cap B = \emptyset \)
\( A \cap C = \emptyset \)
\( B \cap C = \emptyset \)
\( C \cap D = \emptyset \)
\( A \cap B \cap C = \emptyset \)
\( \implies \) Therefore, the pairs (A and B), (A and C), (B and C), (C and D) are mutually exclusive. Also, events A, B, and C together are mutually exclusive.
2. Simple events:
Simple events are those with only one outcome.
\( \implies \) Events A and C are simple events, as \( A = \{\text{HHH}\} \) and \( C = \{\text{TTT}\} \).
3. Compound events:
Compound events are those with more than one outcome.
\( \implies \) Events B and D are compound events, as \( B = \{\text{HHT, HTH, THH}\} \) and \( D = \{\text{HHH, HHT, HTH, HTT}\} \).
In simple words: First, write down all the possible results when you toss three coins. Then, list the results for each event (A, B, C, D). If events have no shared results, they are mutually exclusive. If an event has only one result, it's simple. If it has many results, it's compound.

Exam Tip: Remember these definitions: A simple event has exactly one outcome, a compound event has more than one outcome, and mutually exclusive events have no outcomes in common (their intersection is an empty set).

Question 5. Three coins are tossed. Describe:
1. two events which are mutually exclusive.
2. three events which are mutually exclusive and exhaustive.
3. two events which are not mutually exclusive.
4. two events which are mutually exclusive but not exhaustive.
5. three events which are mutually exclusive but not exhaustive.
Answer: When three coins are tossed, the sample space S is:
\( S = \{\text{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}\} \).
1. Two events which are mutually exclusive:
Let A be the event of "getting three heads" \( A = \{\text{HHH}\} \).
Let B be the event of "getting three tails" \( B = \{\text{TTT}\} \).
These events are mutually exclusive because \( A \cap B = \emptyset \).
2. Three events which are mutually exclusive and exhaustive:
Let A be the event of "getting at most one head" \( A = \{\text{TTT, HTT, THT, TTH}\} \).
Let B be the event of "getting exactly two heads" \( B = \{\text{HHT, HTH, THH}\} \).
Let C be the event of "getting exactly three heads" \( C = \{\text{HHH}\} \).
These three events are mutually exclusive (no common outcomes), and their union \( A \cup B \cup C = S \) covers all possible outcomes, making them exhaustive.
3. Two events which are not mutually exclusive:
Let A be the event of "getting at most two tails" \( A = \{\text{HHH, HHT, HTH, THH, HTT, THT, TTH}\} \).
Let B be the event of "getting exactly two heads" \( B = \{\text{HHT, HTH, THH}\} \).
These events are not mutually exclusive because \( A \cap B = \{\text{HHT, HTH, THH}\} \neq \emptyset \).
4. Two events which are mutually exclusive but not exhaustive:
Let A be the event of "getting exactly one head" \( A = \{\text{HTT, THT, TTH}\} \).
Let B be the event of "getting exactly three tails" \( B = \{\text{TTT}\} \).
These events are mutually exclusive because \( A \cap B = \emptyset \). However, their union \( A \cup B = \{\text{HTT, THT, TTH, TTT}\} \) does not cover all outcomes in S, so they are not exhaustive.
5. Three events which are mutually exclusive but not exhaustive:
Let A be the event of "getting exactly one tail" \( A = \{\text{HHT, HTH, THH}\} \).
Let B be the event of "getting exactly two tails" \( B = \{\text{HTT, THT, TTH}\} \).
Let C be the event of "getting exactly three tails" \( C = \{\text{TTT}\} \).
These three events are mutually exclusive as they have no common outcomes. However, their union \( A \cup B \cup C = \{\text{HHT, HTH, THH, HTT, THT, TTH, TTT}\} \) does not include the outcome HHH, so they are not exhaustive.
In simple words: For each part, we need to pick specific groups of coin toss results that fit the given conditions. Mutually exclusive means no shared results. Exhaustive means all possible results are covered. We show examples for events that meet these criteria or fail to meet them.

Exam Tip: When describing events, clearly state the outcomes included in each set. This makes it easier to perform set operations and demonstrate properties like mutual exclusivity or exhaustiveness.

Question 6. 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 of the dice < 5, Describe the events:
(i) A'
(ii) not B
(iii) A or B
(iv) A and B
(v) A but not C
(vi) B or C
(vii) B and C
(viii) A ∩ B' ∩ C'
Answer: When two dice are thrown, there are \( 6 \times 6 = 36 \) possible outcomes. The sample space S is:
\( S = \{(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), \)
\( (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), \)
\( (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), \)
\( (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), \)
\( (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), \)
\( (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)\} \).
Event A, getting an even number on the first die, includes:
\( A = \{(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), \)
\( (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), \)
\( (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)\} \).
Event B, getting an odd number on the first die, includes:
\( B = \{(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), \)
\( (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), \)
\( (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6)\} \).
Event C, getting a sum of numbers on the dice less than or equal to 5, includes:
\( C = \{(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (4, 1)\} \).
Now, let's describe the requested events:
(i) A' (complement of A): This means not getting an even number on the first die, which is the same as getting an odd number on the first die.
\( \implies A' = B = \{(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6)\} \).
(ii) not B (complement of B): This means not getting an odd number on the first die, which is the same as getting an even number on the first die.
\( \implies \text{not } B = B' = A = \{(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)\} \).
(iii) A or B (union of A and B): This includes outcomes that are in A, or in B, or in both. Since A and B cover all outcomes (first die is either even or odd), their union is the entire sample space S.
\( \implies A \cup B = S = \{(1, 1), (1, 2), ..., (6, 6)\} \).
(iv) A and B (intersection of A and B): This includes outcomes that are in both A and B. Since a number cannot be both even and odd, there are no common outcomes.
\( \implies A \cap B = \emptyset \).
(v) A but not C (difference A - C): This includes outcomes that are in A but not in C.
\( A - C = \{(2, 4), (2, 5), (2, 6), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)\} \).
(vi) B or C (union of B and C): This includes outcomes that are in B, or in C, or in both.
\( B \cup C = \{(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (4, 1), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6)\} \).
(vii) B and C (intersection of B and C): This includes outcomes that are common to both B and C.
\( B \cap C = \{(1, 1), (1, 2), (1, 3), (1, 4), (3, 1), (3, 2)\} \).
(viii) A ∩ B' ∩ C': First, we know \( B' = A \). So, the expression becomes \( A \cap A \cap C' = A \cap C' \).
\( C' \) is the complement of C, meaning outcomes where the sum of numbers on the dice is greater than 5.
\( A \cap C' = \{(2, 4), (2, 5), (2, 6), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)\} \).
In simple words: First, list all 36 possible results when rolling two dice. Then, list the specific results for events A, B, and C. For each question part, apply the set operations like complement (A'), union (or), and intersection (and) to find the elements in the new event.

Exam Tip: Be meticulous when listing outcomes for two dice, as one missing pair can affect all subsequent calculations. Use a table or grid to organize the sample space for clarity.

Question 7. Refer to question 6 above, state true or false:
(i) A and B are mutually exclusive.
(ii) A and B are mutually exclusive and exhaustive.
(iii) A = B'.
(iv) A and C are mutually exclusive.
(v) A and B'are mutually exclusive.
(vi) A', B', C are mutually exclusive and exhaustive.
Answer: We refer to the definitions of events A, B, and C from Question 6.
(i) A and B are mutually exclusive.
This statement is True. Event A means an even number on the first die, and event B means an odd number on the first die. These two events share no common outcomes, as \( A \cap B = \emptyset \).
(ii) A and B are mutually exclusive and exhaustive.
This statement is True. As established above, A and B are mutually exclusive. Also, their union \( A \cup B \) covers all possible outcomes of throwing two dice, representing the entire sample space S.
\( \implies \) Thus, \( A \cup B = S \), making A and B exhaustive events.
(iii) A = B'.
This statement is True. The complement of B (B') means not getting an odd number on the first die, which is the same as getting an even number on the first die. This is exactly event A.
\( \implies \) Therefore, \( A = B' \).
(iv) A and C are mutually exclusive.
This statement is False. The intersection of A and C is \( A \cap C = \{(2, 1), (2, 2), (2, 3), (4, 1)\} \).
\( \implies \) Since this intersection is not empty, \( A \cap C \neq \emptyset \), meaning A and C are not mutually exclusive.
(v) A and B' are mutually exclusive.
This statement is False. Since \( B' = A \) (as shown in iii), then the intersection \( A \cap B' = A \cap A = A \).
\( \implies \) As A is not an empty set, \( A \neq \emptyset \), so A and B' are not mutually exclusive.
(vi) A', B', C are mutually exclusive and exhaustive.
This statement is False. Events A' and B' are mutually exclusive because \( A' \cap B' = \emptyset \). However, when considering A', B', and C together, we see that \( A' \cap C = B \cap C = \{(1, 1), (1, 2), (1, 3), (1, 4), (3, 1), (3, 2)\} \neq \emptyset \). Also, \( B' \cap C = A \cap C = \{(2, 1), (2, 2), (2, 3), (4, 1)\} \neq \emptyset \).
\( \implies \) This means A', B', and C are not mutually exclusive events, nor are they exhaustive (as their union does not cover the entire sample space S).
In simple words: Use the definitions of events A, B, and C from the previous question. For each statement, decide if it's true or false by checking the conditions for mutual exclusivity (no shared outcomes), exhaustiveness (cover all outcomes), or equality of sets.

Exam Tip: To assess "true or false" questions on event relationships, always refer back to the precise definitions of the events and their corresponding sets. Counter-examples (even one shared element for non-mutual exclusivity) are sufficient to prove a statement false.