Give a real-life example of the following: Mutually exclusive and exhaustive Events.

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> [!TLDR]
> A pair of events can be both mutually exclusive (no overlap) and exhaustive (covers everything) at once — shown on a single die roll.

> [!FORMULA]
> Events $A,B$ on sample space $S$ are **mutually exclusive and exhaustive** if both hold together:
> $$A\cap B=\varnothing \qquad	ext{and}\qquad A\cup B=S$$

1. **Set up the experiment.** Roll a single fair die once. Sample space $S=\{1,2,3,4,5,6\}$, so $n(S)=6$.
2. **Define the events.** Let $A=$ "number $\le 3$" $=\{1,2,3\}$ and $B=$ "number $\ge 4$" $=\{4,5,6\}$.
3. **Check mutual exclusivity.**
 $$A\cap B=\{1,2,3\}\cap\{4,5,6\}=\varnothing$$
 No outcome can be both $\le 3$ and $\ge 4$ at once, so $A$ and $B$ are mutually exclusive.
4. **Check exhaustiveness.**
 $$A\cup B=\{1,2,3\}\cup\{4,5,6\}=\{1,2,3,4,5,6\}=S$$
 Every possible face is covered, so $A$ and $B$ are exhaustive. …

Random Experiment	Sample Space
Toss a coin	$S = \{H, T\}$
Roll a die	$S = \{1, 2, 3, 4, 5, 6\}$
Toss two coins	$S = \{HH, HT, TH, TT\}$
Roll two dice	$S = \{(1,1), (1,2), \dots, (6,6)\}$ — 36 outcomes
Draw a card from a deck	$S = \{\text{A♠}, \text{K♠}, \dots, \text{2♣}\}$ — 52 outcomes

Q.Give a real-life example of the following: Mutually exclusive and exhaustive Events.

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Q.Give a real-life example of the following: Mutually exclusive and exhaustive Events.

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