In discrete probability, we assume a well-defined experiment, such as flipping a coin or rolling a die. Each individual result which could occur is called an outcome. The set of all outcomes is called the sample space, and any subset of the sample space is called an event. 

For example, consider the experiment of flipping a coin two times. There are four individual outcomes, namely $HH, HT, TH, TT.$ The sample space is thus $\{HH, HT, TH, TT\}.$ The event "at least one heads occurs" would be the set $\{HH, HT, TH\}.$ If the coin were a normal coin, we would assign the probability of $1/4$ to each outcome. 

In probability theory, the probability $P$ of some event $E$, denoted $P(E)$, is usually defined in such a way that $P$ satisfies a number of axioms, or rules. The most basic and most important rules are listed below.

Probability Rules

1. Probability is a number. It is always greater than or equal to zero, and less than or equal to one. This can be written as $0 \leq P(A) \leq 1$.  An impossible event, or an event that never occurs, has a probability of $0$. An event that always occurs has a probability of $1$. An event with a probability of $0.5$ will occur half of the time. 
2. The sum of the probabilities of all possibilities must equal $1$. Some outcome must occur on every trial, and the sum of all probabilities is 100%, or in this case, $1$. This can be written as $P(S) = 1$, where $S$ represents the entire sample space.
3. If two events have no outcomes in common, the probability that one or the other occurs is the sum of their individual probabilities. If one event occurs in $30\%$ of the trials, a different event occurs in $20\%$ of the trials, and the two cannot occur together (if they are disjoint), then the probability that one or the other occurs is $30\% + 20\% = 50\%$. This is sometimes referred to as the addition rule, and can be simplified with the following: $P(A \ \text{or} \ B) = P(A)+P(B)$. The word "or" means the same thing in mathematics as the union, which uses the following symbol: $\cup $. Thus when $A$ and $B$ are disjoint, we have $P(A \cup B) = P(A)+P(B)$.
4. The probability that an event does not occur is $1$ minus the probability that the event does occur. If an event occurs in $60\%$ of all trials, it fails to occur in the other $40\%$, because $100\% - 60\% = 40\%$. The probability that an event occurs and the probability that it does not occur always add up to $100\%$, or $1$. These events are called complementary events, and this rule is sometimes called the complement rule. It can be simplified with $P(A^c) = 1-P(A)$, where $A^c$ is the complement of $A$.
5. Two events $A$ and $B$ are independent if knowing that one occurs does not change the probability that the other occurs. This is often called the multiplication rule. If $A$ and $B$ are independent, then $P(A \ \text{and} \ B) = P(A)P(B)$. The word "and" in mathematics means the same thing in mathematics as the intersection, which uses the following symbol: $\cap$. Therefore when A and B are independent, we have $P(A \cap B) = P(A)P(B).$

Extension of the Example

Elaborating on our example above of flipping two coins, assign the probability $1/4$ to each of the $4$ outcomes. We consider each of the five rules above in the context of this example.

1. Note that each probability is $1/4$, which is between $0$ and $1$.

2. Note that the sum of all the probabilities is $1$, since  $\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}=1 $.

3. Suppose $A$ is the event exactly one head occurs, and $B$ is the event exactly two tails occur. Then $A=\{HT,TH\}$ and $B=\{TT\}$ are disjoint. Also, $P(A \cup B) = \frac{3}{4} = \frac{2}{4}+\frac{1}{4}=P(A) + P(B).$

4. The probability that no heads occurs is $1/4$, which is equal to  $1-3/4$. So if $A=\{HT, TH, HH\}$ is the event that a head occurs, we have $P(A^c)=\frac{1}{4}=1 - \frac{3}{4}=1-P(A).$

5. If $A$ is the event that the first flip is a heads and $B$ is the event that the second flip is a heads, then $A$ and $B$ are independent. We have $A=\{HT,HH\}$ and $B=\{TH,HH\}$ and $A \cap B = \{HH\}.$ Note that $P(A \cap B) = \frac{1}{4} =\frac{1}{2}\cdot \frac{1}{2} = P(A)P(B).$

Die

Dice are often used when learning the rules of probability.