Sample space, events and axioms of probability
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Sample space and sample points
A sample space, often denoted by the letter
Sample space of tossing a coin
What is the sample space of tossing a coin:
once?
twice?
Solution. There are two outcomes when a coin is tossed: heads (
Sample space of rolling a dice
What is the sample space of rolling a dice? How many sample points are there?
Solution. The sample space of rolling a dice is:
The number of sample points is
Events, simple events and compound events
An event is a set of outcomes of an experiment. Events are always a subset of the sample space. A simple event has only one outcome, that is, it can only happen in one way. A compound event has two or more outcomes and it can be decomposed into simple events.
Events of rolling a dice
What are some events associated with a single roll of a dice?
Solution. Here is a list of some events that may be of interest:
event
- observe an odd number, that is, .event
- observe a number greater than , that is, .event
- observe a , that is, .event
- observe a , that is, .
Here, event
On the other hand, events
simple event
- observe a .simple event
- observe a .and so on.
Notice that any event, whether it be compound or simple, is a subset of the sample space:
Just like for sets, we can visualize events using a Venn diagram:

Recall that the compound event
We can visualize event

Tossing a coin
Suppose we toss a coin three times. What are some examples of events, simple events and compound events?
Solution. Some examples of events we might be interested in are:
event
- getting all heads.event
- getting exactly one tails.event
- getting more than one tails.event
- getting at least two heads.
Note the following:
event
is a simple event because there is only one way in which this event occurs .event
is a compound event because this outcome can occur in multiple ways , or .events
and are also compound events. For instance, event occurs when we either get two or three tails.
Mutually exclusive events
If two events are mutually exclusive or disjoint, then they cannot happen simultaneously.
Mutually exclusive events of rolling a dice
Suppose we roll a dice once. Let's define the following events:
event
- getting an even number.event
- getting an odd number.
These events are mutually exclusive because they cannot occur at the same time.
All simple events are mutually exclusive
Simple events are those that have only one outcome, that is, they can only happen in one way. For example, the simple events of tossing a coin are:
event
- getting a heads.event
- getting a tails.
Since we cannot get a heads and tails at the same time, these simple events must be mutually exclusive.
Formal definition of probability based on axioms
Let
Axiom 1:
Axiom 2:
Axiom 3: If
Intuition. The first two axioms of probability are straight-forward. Axiom 1 states that the probability of an event cannot be negative. If
Axiom 2 states that the probability of the sample space
Axiom 3 is much less intuitive. This axiom states that if events are disjoint or mutually exclusive, then the probability of their union must be equal to the sum of the probabilities of each event. For example, suppose we roll a fair dice once. The sample space is
event
- observe a .event
- observe a .
These events are mutually exclusive, that is, they cannot happen at the same time. Axiom 3 states that the probability that either events
Let's understand what this intuitively means. For a fair dice, events
What we have done here is applied the third axiom of probability! Similarly, we can use the third axiom for calculating the probability of rolling an odd number:
In this way, all three axioms of probability make a lot of sense!
Independent events
Two events are said to be independent if the occurrence (or non-occurrence) of one event does not affect the probability that the other event will occur.
Coins toss
Suppose we toss a coin twice and we define the following events:
event
- obtaining a heads for the first toss.event
- obtaining a heads for the second toss.
Because the outcome of the first coin toss does not affect the outcome of the next coin toss, these events are independent.