Download Sample Space Ω: The set of all possible outcomes in an experiment

Sample Space Ω: The set of all possible outcomes in an experiment.
The sample space must be
-Mutually exclusive: Either one outcome happens or another, but not both.
-Collectively exhaustive: All the possible outcomes are in the set.
There are different ways to express the sample space.
Discrete example: Consider flipping a coin twice.
The sample space could be described using set notation.
Let H denote heads when flipped, and T denote tails when flipped.
Ω = {HH, HT, TH, TT}
It might be helpful to think of it as a diagram.
Yet another way could be with a diagram which gives a sequential description.
Event: A subset of the sample space.
-
Probability is assigned to events.
If A is a subset and the outcome is in A, then we say A has occurred.
Axioms of Probability:
1.) Nonnegativity: P(A) ≥ 0
2.) Normalization: P(Ω) = 1
3.) Additivity: If A ∩ B = Ø, then P(A∪B) = P(A) + P(B)
To make sense of axiom 3 we can think in terms of area. For the purposes of having a picture to
visualize our understanding, the probability represents the ratio of how much area the event takes
up to the area of the sample space.
You may be wondering why there is not an axiom stating that P(A) ≤ 1.
The reason is that we can derive that fact from our axioms.
Note: The compliment of A, denoted by A’, is all of the elements in the sample space which are
not in A.
By the normalization,
1 = P(Ω)
By definition,
P(Ω) = P(A∪ A’)
By additivity,
P(A ∪ A’) = P(A) + P(A’)
Therefore,
1 = P(A) + P(A’)
Or
P(A’) = 1 – P(A)
Nonnegativity states that P(A’) ≥ 0,
this implies that,
1 – P(A) ≥ 0
So
-P(A) ≥ -1
Or that
P(A) ≤ 1.
We conclude that P(A) ≤ 1.
That is something interesting that came from our axioms.
To continue along that path, let’s suppose we have three sets of elements (i.e. A, B, and C) such
that A ∩ B ∩ C = Ø. What is P(A ∪ B ∪ C)?
P(A ∪ B ∪ C) = P[(A ∪ B) ∪ C]
= P(A ∪ B) + P(C)
= P(A) + P(B) + P(C)
We can see that if A ∩ B ∩ C = Ø, then P(A ∪ B ∪ C) = P(A) + P(B) + P(C).
We could play this game for any number of disjoint sets.
Countable additivity axiom:
If we have a sequence of sets that are disjoint, then their total probability is the sum of their
individual probabilities.
Discrete example
Consider the following diagram where each dot represents a unique outcome; all outcomes are
equally likely.
Here, the sample space Ω has 8 outcomes; three of those outcomes are in A.
The probability of A denoted by P(A) in this example is
3
P(A) = 8
In general, with similar discrete experiments, we have
#𝐴
P(A) = #Ω
Where #A denotes the number of elements in A, and #Ω denotes the number of elements in the
sample space.