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 A1 and A2 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:

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