Properties

The list of possible outcomes of an event is called its sample space. Suppose that you toss a perfectly balanced coin. Then the two outcomes Heads and Tails in the sample space

S={ Heads, Tails }

will occur with the same frequency. In other words, the probability of each event is 1/2. Therefore
P(Heads) = P(Tails) = 1/2 > 0




In general, we can define:

Property 1:

If A is an outcome in a sample space S, then
P(A) > 0
Now consider the coin toss example again:

If the coin is tossed 3 times and Heads occurs every one of those times, then number of times Heads occurs is
n(Heads) = 3
and
P(3 Heads) = ( 1/2 ) 3 = 1/8

The probability of all heads will get smaller and smaller as the number of tosses increases, but it will always be greater than zero.
Thus, we can define the second property as:

Property 2:

If A is an outcome in a sample space S, then
P(A)<= 1
In the coin toss example, the probability of Heads outcome is 1/2 and the probability of the Tails outcome is 1/2.
Therefore,
P(Heads) + P(Tails) = 1/2 + 1/2 = 1
This is one of the basic properties in probability:

Property 3:


If a sample space S contains k outcomes:
S= { A1, A2, A3, ... , Ak}

then
P(A1) + P(A2) + ... + P(Ak) = 1