Information of "event A has high probability value",
"event B has medium probability value",
"event C has low probability value", ...
is called "probability distribution"
==================================================
Elementary event (or atomic event):
Elementary event has one number of sample
$$$P({\text{Diamond}})=0.1$$$
$$$P({\text{Heart}})=0.2$$$
$$$P({\text{Spade}})=0.3$$$
$$$P({\text{Clover}})=0.4$$$
Then, you can calculate probability value of all kinds of events,
which has 2 samples, 3 samples, etc
according to 3rd rule of Kolmogorov's axiom
$$$P({\text{Heart,Spade}})=0.2+0.3=0.5$$$
==================================================
Probability mass function:
Probability mass function defines probability values
to each elementary event,
when there are only finite number of events
==================================================
$$$P(\{1\})=0.2$$$
$$$P()$$$ is probability function
$$$\{1\}$$$ is event which has one sample
$$$0.2$$$ is probability value for event $$$\{1\}$$$
$$$p(1)=0.2$$$
$$$p()$$$ is probability mass function
$$$1$$$ is elementary event which has number 1
$$$0.2$$$ is probability value for elementary event 1
==================================================
$$$P(\{1,2\})=0.2$$$
$$$P()$$$ is probability function
$$$\{1,2\}$$$ is event which has 2 samples
$$$0.2$$$ is probability value for event $$$\{1,2\}$$$
$$$p(1,2)$$$ can be defined
==================================================
Simple interval event A: $$$A=\{a\le x < b\}$$$
$$$A = \{ a \leq x < b \} \;\; \rightarrow \;\; P(A) = P(\{ a \leq x < b \}) = P(a, b)$$$
$$$P(B) = P(\{ -2 \leq x < 1\}) + P(\{2 \leq x < 3\}) = P(-2, 1) + P(2, 3)$$$
$$$P(B) = P(\{ -2 \leq x < 3 \}) - P(\{ 1 \leq x < 2\}) = P(-2, 3) - P(1, 2)$$$
==================================================
Cumulative distribution function:
From above, you used 2 numbers to define "interval" or "simple interval event A"
To use only one number, you can use negative infinity
$$$
S_{-1} = \{ -\infty \leq X < -1 \} \\
S_{0} = \{ -\infty \leq X < 0 \} \\
S_{1} = \{ -\infty \leq X < 1 \} \\
S_{2} = \{ -\infty \leq X < 2 \} \\
\vdots \\
S_{x} = \{ -\infty \leq X < x \}$$$
==================================================
In interval $$$\{ a \leq x < b \}$$$
simple intervale probability $$$P(a, b) = P(-\infty, b) - P(-\infty, a)$$$
==================================================
Probability density function:
derivative of Cumulative distribution function
==================================================
Probability distribution function:
Probability mass function
Cumulative distribution function
Probability density function
==================================================