## Properties Of Probabilities

Besides the fact that a probability must be a value between 0 or 1, the following are additional characteristics of probability.

**If an event is certain to occur, the probability of that event is equal to 1**.

For example, if we have a coin with heads on both sides, then:

$Pr(heads) = 1$

As another example, by definition, a sample space must have a probability equal to one, as it encompasses all possible events. Thus:

$Pr(S) = 1$

Where S stands for Sample Space.

**If an event is impossible, the probability of that event is equal to 0**.

Using the same example as before, if we have a coin with heads on both sides then:

$Pr(tails) = 0$

As another example, using the concepts from set theory, we have:

$Pr(\emptyset) = 0$

Where $\emptyset$ stands for empty set, which is a set with no events or outcomes. An empty set really just exists as a theoretical concept to help with explaining other concepts.

**The probability of a union of two disjoint events is equal to the sum of their probabilities**.

Assume A_{1} and A_{2} are two disjoint events. Key emphasis on disjoint.

$Pr(A_1\cup A_2) = Pr(A_1) + Pr(A_2)$

For example, let’s consider rolling a die. Two disjoint events are rolling a 1 and rolling a 6. The probability of rolling a 1 is $1/6$ and the probability of rolling a 6 is $1/6$. Thus:

$Pr(rolling\_a\_1 \cup rolling a 6) = Pr(1) + Pr(6)$

$Pr(rolling a 1 \cup rolling a 6) = 1/6 + 1/6$

$Pr(rolling a 1 \cup rolling a 6) = 1/3$

Remember, disjoint events (in some texts referred to as mutually exclusive events) are events that cannot occur at the same time. In the above, with one roll of a die, 1 and 6 cannot both occur. Now we extend the concept further.

**The sum of an infinite sequence of disjoint events is just the sum of their probabilities**.

This is the same rationale as above except a bit more theoretical. To notate this, we have the following, where $A_i$ are i disjoint events.

$Pr(\bigcup_{i=1}^\infty A_i) = \sum_{i=1}^\infty Pr(A_i)$

The above may look confusing so let’s break it down. $\bigcup A_i$ means the union of i disjoint events. The subscript i in $A_i$ is just an index value. i ranges from 1 to $\infty$. We know this because below $\bigcup_{i=1}$ we have i =1, meaning that i starts at 1. At the top of $\bigcup^\infty$ we have $\infty$, which tells you the upper bound of where i ranges to. The same concept applies to the right hand side. $\sum$ is the symbol for summation, and we sum everything to the right of it, which in this case is $Pr(A_i)$. As you know, there are an infinite number of $Pr(A_i)$ to sum together. Another way to think of the above, if it is still confusing is:

$Pr(A_1 \cup A_2 \cup \dots \cup A_\infty) = Pr(A_1) + Pr(A_2) + \dots + Pr(A_\infty)$