Introduction to Probability
Introduction
Now that you understand the basics of sets, you'll learn how this knowledge can be used to calculate your first probabilities! In this section, you'll learn how to use sets to create probabilities, and you'll learn about the very foundations of probability through the three probability axioms.
Objectives
You will be able to:
- Learn about experiments, outcomes and event space
- Understand probability through the law of relative frequency
- Learn about the probability axioms
- Learn about the addition law of probability
- Learn that where each outcome is equally likely, the probability is equal to number of outcomes in the event space divided by number of outcomes in the sample space
Experiment and outcomes
Previously, we defined sets and related concepts. Now let's look at the set
In this context, we'd call throwing the dice a random experiment. The result of this experiment is then called the outcome. Note that
In the context of experiments, we denote
Other examples of sample spaces:
- Number of text messages a day:
$S = {x \mid x \in \mathbb{Z}, x \geq 0}$ - Hours of TV a day:
$S = {x \mid x \in \mathbb{R}, 0 \leq x \leq 24 }$
Event space
In this context, we can also define the event space. The event space is a subset of the sample space,
For example, the event "throwing a number higher than 4" would result in an event space
Summarized, the event space is a collection of events that we care about. We say that event
Using the concepts of sample space and event space, we can now introduce the concept of probability.
Other examples of event spaces based on previously defined sample spaces:
- Low text message volume day:
$E = {x \mid x \in \mathbb{Z}, 0 \leq x \leq 20 }$ - Bingewatch day:
$E = {x \mid x \in \mathbb{R}, x \geq 6 }$
Introduction to probability
The law of relative frequency
While conducting an endless stream of experiments, the relative frequency by which an event will happen becomes a fixed number.
Let's denote an event
This is a useful definition, but in practice, it is fairly cumbersome as
To understand this even better, let's look at another way of writing this. Again, if we say each outcome in
where
The problem here is, however, what if all the singular events are not equally likely?
That's why in the early 20th century, Kolmogorov and Von Mises came up with 3 axioms that altogether are equivalent to the law of relative frequency.
Probability axioms
The three axioms are
1. Positivity
A probability is always bigger than or equal to 0, or
2. Probability of a certain event
If the event of interest is the sample space, we say that the outcome is a certain event, or
3. Additivity
The probability of the union of 2 exclusive events is equal to the sum of the probabilities of the individual events happening.
If
Addition law of probability
The additivity axiom is great, but most of the time events are not exclusive. A very important proberty is the addition law or probability or the sum rule.
Put in words, the probability that
Summary
Well done! In this section, you learned how to use sets to get to probabilities. You learned about experiments, event spaces and outcomes. Next, you learned about the law of relative frequency and how it can be used to calculate probabilities, along with the three probability axioms.