Probabilistic sample(or random sample, sample):
One realizable phenomenon from the probabilistic problem you want to solve
Or one sampled case
Sample space $$$\Omega$$$:
Set which contains all possible samples
Task of defining sample sapce:
Define which phenomenon is possible to occur
and which phenomenon is impossible to occur
==================================================
Sample space when you toss the coin
$$$\Omega =\{H,T\}$$$
==================================================
Some cases has set of entire real numbers as sample space
$$$\Omega = \mathbf{R}$$$
==================================================
Possible events: possible sub sets of sample space $$$\Omega$$$
Sample space $$$\Omega=\{H,T\}$$$
Possible events: $$$\phi,\{H\},\{T\},\{H,T\}$$$
==================================================
Probability is function which takes all events
and which outputs number
$$$P(A)=0.1$$$
P() is function P
A is event
0.1 is probability value
==================================================
Kolmogorov's axioms
1. $$$P(\text{all\_events})\ge 0$$$
2. $$$P(\Omega)=1$$$
3. If $$$A\cap B=\phi$$$, then $$$P(A\cup B)=P(A)+P(B)$$$
==================================================
Interpretion about probability value:
1. Frequentist
2. Baysian
==================================================