Q.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

Concept: Set Operations — classifying events based on whether they can occur together (mutually exclusive) and how many outcomes they contain (simple vs. compound).

Step 1: List the sample space.

$S = \{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT\}$.

Step 2: Write each event as a set.

• $A = \{HHH\}$
• $B = \{HHT, HTH, THH\}$
• $C = \{TTT\}$
• $D = \{HHH, HHT, HTH, HTT\}$

Step 3: Check mutual exclusivity.

Two events are mutually exclusive if they have no common outcome.

• $A$ and $C$: disjoint.
• $A$ and $B$: disjoint.
• $B$ and $C$: disjoint.
• $A$ and $D$: share $HHH$ → not mutually exclusive.
• $B$ and $D$: share $HHT, HTH$ → not mutually exclusive.
• $C$ and $D$: disjoint.

Step 4: Identify simple vs. compound. …

When three coins are tossed, the sample space has 8 equally likely outcomes:

$S = \{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT\}$.

Let’s list each event explicitly:

• A: 'three heads show' → $\{HHH\}$
• B: 'two heads and one tail show' → $\{HHT, HTH, THH\}$
• C: 'three tails show' → $\{TTT\}$
• D: 'a head shows on the first coin' → $\{HHH, HHT, HTH, HTT\}$

Now we classify them.

(i) Mutually exclusive events

Two events are mutually exclusive if they have no outcome in common — they cannot happen at the same time.

Check each pair:

1. A and C: $A \cap C = \{HHH\} \cap \{TTT\} = \emptyset$ → mutually exclusive.
2. A and B: $A \cap B = \{HHH\} \cap \{HHT, HTH, THH\} = \emptyset$ → mutually exclusive.
3. A and D: $A \cap D = \{HHH\} \cap \{HHH, HHT, HTH, HTT\} = \{HHH\} \neq \emptyset$ → not mutually exclusive.
4. B and C: $B \cap C = \{HHT, HTH, THH\} \cap \{TTT\} = \emptyset$ → mutually exclusive.
5. B and D: $B \cap D = \{HHT, HTH, THH\} \cap \{HHH, HHT, HTH, HTT\} = \{HHT, HTH\} \neq \emptyset$ → not mutually exclusive.
6. C and D: $C \cap D = \{TTT\} \cap \{HHH, HHT, HTH, HTT\} = \emptyset$ → mutually exclusive.

So the mutually exclusive pairs are: (A, C), (A, B), (B, C), (C, D).

(ii) Simple events

A simple event contains exactly one outcome from the sample space. …