Q.A coin is tossed three times, consider the following events. A: 'No head appears', B: 'Exactly one head appears' and C: 'Atleast two heads appear'. Do they form a set of mutually exclusive and exhaustive events?

Concept: Mutually exclusive and exhaustive events

The sample space when a coin is tossed three times contains 8 outcomes:

Now identify each event:

• Event $A$ (no head): $\{\text{TTT}\}$ — 1 outcome
• Event $B$ (exactly one head): $\{\text{HTT, THT, TTH}\}$ — 3 outcomes
• Event $C$ (at least two heads): $\{\text{HHT, HTH, THH, HHH}\}$ — 4 outcomes

Check mutual exclusivity: No outcome appears in more than one event. For instance, TTT is only in $A$, HTT is only in $B$, and HHH is only in $C$. So $A \cap B = \phi$, $B \cap C = \phi$, $A \cap C = \phi$. …

Understanding Mutually Exclusive and Exhaustive Events

When we toss a coin three times, every outcome is a sequence of heads and tails. The question asks whether the three events $A$, $B$, and $C$ satisfy two properties:

• Mutually exclusive: No two events share a common outcome. If one happens, the others cannot.
• Exhaustive: Together, the events account for every possible outcome in the sample space.

Think of it as partitioning a set: mutually exclusive means the partitions don't overlap, exhaustive means nothing is left out.

The Sample Space

A coin tossed three times produces $2^3 = 8$ equally likely outcomes:

Now let's identify which outcomes belong to each event.

Step-by-Step Analysis

1. Event $A$: 'No head appears' This means all three tosses are tails.

Number of outcomes: $|A| = 1$.

2. Event $B$: 'Exactly one head appears' Exactly one $H$ and two $T$s. The head can appear in the first, second, or third position.

Number of outcomes: $|B| = 3$.

3. Event $C$: 'At least two heads appear' This means two heads or three heads.

Number of outcomes: $|C| = 4$.

4. Check if they are mutually exclusive

   Two events are mutually exclusive if their intersection is empty.

   • $A \cap B$: Can we have no heads and exactly one head? No. $A \cap B = \emptyset$.
   • $A \cap C$: Can we have no heads and at least two heads? No. $A \cap C = \emptyset$.
   • $B \cap C$: Can we have exactly one head and at least two heads? No. $B \cap C = \emptyset$.

   Since no two events share any outcome, they are mutually exclusive.

5. Check if they are exhaustive …