Sample space and events
Sample space S: Set of ALL possible outcomes.
Example: coin={H,T}.
Die={1,2,3,4,5,6}.
The sample space of two dice (36 equally-likely outcomes): count the favourable squares over 36.
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Event: Any subset of sample space.
Example: rolling even={2,4,6}.
Always count carefully: Sample space must be exhaustive and outcomes equally likely.
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Calculating probabilities
Worked example
Die rolled.
Find P(even), P(>3).
Solution
- S={1,2,3,4,5,6}
- P(even)=3/6=1/2
- P(>3)=3/6=1/2
Final answer
Both are 1/2.
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Theoretical vs empirical probability
| Type | Definition |
|---|---|
| Theoretical | Based on logic (equally likely) |
| Empirical | Relative frequency from trials |
Worked example
Coin: theory P(H)=0.5. 100 flips give 52 heads.
Empirical P(H)?
Solution
- Empirical=52/100=0.52
- More trials: empirical approaches theoretical
Final answer
Empirical=0.52.
Complementary and mutually exclusive
Complementary: Events covering all: P(A)+P(Ac)=1.
Example: heads or tails.
Mutually exclusive: Cannot happen together: P(A and B)=0.
Example: rolling 2 AND 3.
Worked example
P(rain)=0.3.
Find P(no rain).
Solution
- P(no rain)=1-0.3=0.7
- Mutually exclusive AND complementary
Final answer
0.7.