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?

Step 1: Determine the Sample Space

First, we need to list all possible outcomes when a coin is tossed three times. Each toss can result in either a Head (H) or a Tail (T). With three tosses, there are $2^3 = 8$ possible outcomes, which form our sample space $S$.

Step 2: Define Event A: No head appears

Event A is defined as 'No head appears'. This means all three tosses must result in tails. From our sample space, only one outcome satisfies this condition.

Step 3: Define Event B: Exactly one head appears

Event B is defined as 'Exactly one head appears'. This means that out of the three tosses, exactly one is a head and the other two are tails. We identify all such outcomes from the sample space.

Step 4: Define Event C: At least two heads appear

Event C is defined as 'At least two heads appear'. This means there can be two heads or three heads. We list all outcomes from the sample space that have two or three heads.

Step 5: Check for Mutual Exclusivity

Events are mutually exclusive if they cannot occur at the same time, meaning their intersections are empty. We check the intersection of each pair of events: $A \cap B$, $A \cap C$, and $B \cap C$. Since there are no common outcomes between any pair, the events are mutually exclusive.

Step 6: Check for Exhaustiveness

Events are exhaustive if their union covers the entire sample space. We combine all outcomes from events A, B, and C and compare this union to the original sample space $S$. Since $A \cup B \cup C = \{$ TTT, HTT, THT, TTH, HHH, HHT, HTH, THH \}, which is equal to S, the events are exhaustive.

Step 7: Conclusion

Since the events A, B, and C are both mutually exclusive (no overlap) and exhaustive (cover all possible outcomes), they form a set of mutually exclusive and exhaustive events.

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