The set of all possible outcomes of a statistical experiment is called the sample space and is represented by the symbol $S$. Each outcome in a sample space is called an element or member of the sample space or simply a sample point.

A sample space must be non-empty.

An event is a subset of a sample space. For example, we can say the event $A$ that the roll of a die is odd is $A=\{1,3,5\},$ the event $B$ that the roll of a die is even is $B=\{2,4,6\}$, and that the event that the roll of a die is less than or equal to 4 is $C=\{1,2,3,4\}.$

Note: This applies to sample spaces of discrete events and is a bit hand wavy.

We can assign a probability or weight to each sample point in a sample space by giving it a value ranging from 0 to 1. Events that are more likely to occur have a probability closer to 1, and events that are less likely to occur have a probability closer to 0. We give a probability to all sample points in a sample space such that the sum of the probabilities of all sample points in a sample space is 1.

Then, the probability of an event $A$ is the sum of the probabilities of all sample points in $A.$ Therefore,

$$0\le P(A)\le 1,P(\mathrm{\varnothing })=0,P(S)=1.$$

The odds of an event $A$ occurring are defined as

$$\mathrm{odds}(A)=\frac{P(A)}{P(A^c)}=\frac{P(A)}{1-P(A)}$$

that is, the probability $A$ occurs divided by the probability of the complement of $A$ (i.e., the probability that $A$ does not occur.)

A log probability is just a logarithm of a probability.

The events $A$ and $B$ are said to be independent if

$$P(A\cap B)=P(A)P(B).$$

From this, we can give the formula for conditional probability. The probability of $A$ given $B$ is

$$P(A|B)=\frac{P(A\text{and}B)}{P(B)}.$$

That is to say, the probability of $A$ occurring given $B$ has occurred is the portion of times $B$ occurs that $A$ also occurs.

If $P(A)=P(A|B)$, then $A$ and $B$ are independent events, and $P(A)\cdot P(B)=P(A\text{and}B).$

Let $A,B$ and $C$ be events. $A$ and $B$ are said to be conditionally independent given $C$ if and only if and $P(A|B,C)=P(A|C).$

Equivalently, conditional independence may be stated as $P(A,B|C)=P(A|C)P(B|C),$ where $P(A,B|C)$ is the joint probability of $A$ and $B$ given $C.$