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- Sample space: the set which contains all possible events.
- Example:
* Trial: toss a coin one time.
* Sample space $$$S=\{H,T\}$$$
* H, T from S: sample points
* Possible events (subset of S):
$$$\phi$$$ (event where you get nothing), {H} (event where you get H), {T} (event where you get T), {H,T} (event you get H or T)
* {H}, {T} from all possible events: fundamental events which composed of sample points
* {H,T}: event, fundamental events $$$\in$$$ event
* Probability: function, domain is all possible events, range is numbers
$$$\text{domain: all possible events} \xrightarrow {\text{Probability as function}} \text{numbers as probability value}$$$
This is notes which I wrote as I take following lecture
http://www.kocw.net/home/search/kemView.do?kemId=1189957
- Probability_space_Axioms_on_probability
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If sample space S is mapped to probability values, sample space S is mapped to probability space.
Each event has probability value in probability space.
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Axioms on probability
- $$$0 \le P[A_i]$$$
$$$A_i$$$: ith event
$$$P[A_i]$$$: probability value of ith event A occuring
- $$$\sum\limits_{i=1}^n P[A_i]=1$$$
- If $$$A_i \cap A_j = \phi$$$, then, $$$P[A_i \cup A_j] = P[A_i] + P[A_j]$$$
For example, $$$P(\{H \cup T\})=P(\{H\})+P(\{T\})$$$
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Expanded characteristics of probability from axioms
- $$$P[A^c]=1-P[A]$$$
- $$$P[A] \le 1$$$
- $$$P[\phi] = 0$$$
- When $$$\{A_1,A_2,\cdots,A_N\}$$$ is given, if $$${A_i\cap A_j=\phi}$$$ is true,
then, $$$P[\cup_{k=1}^N A_k]=\sum\limits_{k=1}^N P[A_k]$$$
- $$$P[A_1\cup A_2]=P[A_1]+P[A_2]-P[A_1\cap A_2]$$$
- $$$P[\cup_{k=1}^N A_k]=\sum\limits_{k=1}^N P[A_k] - \sum\limits_{j\lt k}^N P[A_k\cap A_j]+\cdots+(-1)^{N+1} P[A_1\cap A_2 \cap \cdots \cap A_N]$$$
- If $$$A_1 \subset A_2$$$ is true, then, $$$P[A_1] \le P[A_2]$$$