Let’s explain each term clearly using the example of tossing a coin and a die together:

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(i) Sample Space

The sample space is the set of all possible outcomes in an experiment.

Example from this experiment: When a coin and a die are tossed together:

• The coin can result in either Heads (H) or Tails (T).
• The die can result in one of six numbers: 1, 2, 3, 4, 5, or 6.

Thus, the sample space $S$ for this experiment is: $S = \{(H, 1), (H, 2), (H, 3), (H, 4), (H, 5), (H, 6), (T, 1), (T, 2), (T, 3), (T, 4), (T, 5), (T, 6)\}.$ It contains 12 possible outcomes.

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(ii) Sample Point

A sample point is a single specific outcome from the sample space.

Example from this experiment: If the coin lands on Heads (H) and the die shows 3, the sample point is: $(H, 3).$ Each pair such as $(H, 1)$, $(T, 4)$, etc., is a sample point.

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(iii) An Event

An event is a subset of the sample space. It includes one or more outcomes.

Example from this experiment: Let’s consider the event $A$: "The die shows an even number." The outcomes in this event are: $A = \{(H, 2), (H, 4), (H, 6), (T, 2), (T, 4), (T, 6)\}.$ This is a subset of the sample space.

Another example: Event $B$: "The coin shows Tails." The outcomes in this event are: $B = \{(T, 1), (T, 2), (T, 3), (T, 4), (T, 5), (T, 6)\}.$

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(iv) A Solitary Event

A solitary event is an event consisting of exactly one outcome (a single sample point).

Example from this experiment: Let’s consider the event $C$: "The coin shows Heads, and the die shows 5." This event has only one outcome: $C = \{(H, 5)\}.$

Another example: The event $D$: "The coin shows Tails, and the die shows 1," is also a solitary event: $D = \{(T, 1)\}.$

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Summary Table

Term	Definition	Example from the experiment
Sample Space	All possible outcomes	$S = \{(H, 1), (H, 2), \dots, (T, 6)\}$
Sample Point	A single specific outcome	$(H, 3)$
Event	A subset of the sample space	$A = \{(H, 2), (T, 4), \dots\}$
Solitary Event	An event with one outcome	$C = \{(H, 5)\}$

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Questions to Expand:

1. How is the concept of sample space applied to more complex experiments like rolling two dice?
2. Can an event include the entire sample space? What would it represent?
3. How does the probability of a solitary event compare with larger events?
4. If the coin is biased, how would it affect the probabilities of events in this experiment?
5. Can there be overlapping events in this sample space? Give examples.

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Tip: Always identify the sample space first before working with probabilities, as it is the foundation for solving all probability-related problems!