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 foundations of probability through the three probability axioms.
You will be able to:
- Compare experiments, outcomes, and the event space
- Calculate probabilities by using relative frequency of outcomes to event space
- Describe the three axioms of probability
- Describe the addition law of probability
For the following examples, we will consider what happens when throwing a single 6-sided die ("die" is the singular of "dice").
When you throw a die once, you can consider this a random experiment. The result of this "experiment" is the outcome. So, for example, the outcome could be a 2.
An event is the outcome of a particular random experiment. So, for example, "rolling the die and getting a 5" is an event.
The sample space represents the universe of all possible outcomes. With our die rolling example, that space includes the values of 1, 2, 3, 4, 5, and 6.
The event space is a subset of the sample space. You can think of it as the collection of events we "care about" out of all possible events. For example, our event space could be "rolling a number higher than a 4", which would include the values of 5 and 6.
As you may have noticed, we used the term "subset" to describe event spaces. Sets are a very useful way to represent the probability concepts described above. Let's expand on that now.
Let's define the sample space for rolling a single die as the set
You can then say that:
-
$S$ defines all the possible outcomes when throwing the die once -
$S$ is our universal set$\Omega$ , as seen before
Other examples of sample spaces:
- In this case,
$S$ is equal to some number x, with x being a non-negative integer - Mathematically, x being an integer looks like
$x \in \mathbb{Z}$ .$\mathbb{Z}$ is a "special" set, containing all integers. - x being non-negative looks like
$x \geq 0$ - To represent the set of x values that meet both requirements, we'll use an additional way of defining a set, called the "set builder" notation. In set builder notation, the vertical bar
$\mid$ means "such that", and the conditions are separated by commas - Our overall definition of
$S$ is$S = {x \mid x \in \mathbb{Z}, x \geq 0}$ . In other words,$S$ contains all instances of x such that x is an integer and x is greater than or equal to zero.
- In this case, let's say that x is a real number between 0 and 24
- Mathematically, x being a real number (roughly equivalent to a floating point value, although there are subtle differences) looks like
$x \in \mathbb{R}$ .$\mathbb{R}$ is another special set, the set of all real numbers. - x being between 0 and 24 looks like
$0 \leq x \leq 24$ - Putting that all together, we get
$S = {x \mid x \in \mathbb{R}, 0 \leq x \leq 24 }$ . In other words,$S$ contains all instances of x such that x is a real number and x is between 0 and 24.
Let's define the event space as
An example of
Or if
Once
Other examples of event spaces based on previously defined sample spaces:
- We can define this as 20 or fewer text messages
- The event space still includes only non-negative integers, but this time there is an upper bound of 20 also
-
$E = {x \mid x \in \mathbb{Z}, 0 \leq x \leq 20 }$ . In other words,$S$ contains all instances of x such that x is an integer between 0 and 20.
- We can define this as 6 or more hours watched
- The event space still includes only real numbers below 24, but now the lower bound is 6 rather than 0
-
$E = {x \mid x \in \mathbb{R}, 6 \leq x \leq 24 }$ . In other words,$S$ contains all instances of x such that x is a real number between 6 and 24.
Once you understand sample spaces and event spaces, you understand the foundational concepts of probability.
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 by
In other words, the probability of the event ($P(E)$) equals the limit as the number of experiments
This is the basis of a frequentist statistical interpretation: an event's probability is the ratio of the positive trials to the total number of trials as we repeat the process infinitely.
In the early 20th century, Kolmogorov and Von Mises came up with three axioms that further expand on the idea of probability. The three axioms are:
A probability is always bigger than or equal to 0, or
If the event of interest is the sample space (
The probability of the union of two exclusive events is equal to the sum of the probabilities of the individual events happening.
Remember the inclusion-exclusion principle states that:
If we know that
The same logic works for the probability of events in two event space sets. If the events are exclusive โ they never happen at the same time, so the intersection between them is empty โ you can simply add the two probabilities together.
If
The additivity axiom is great, but most of the time events are not exclusive. Then we need to bring in the rest of the inclusion-exclusion principle (subtracting the intersection), which is now referred to as the addition law of probability or the sum rule when we are talking about probabilities.
Put in words, the probability that
Let's reconsider the dice example to explain what was explained before:
Let's consider two events: event
"what is the probability that your outcome will be a 6, or an odd number?"
There is a 2/3 probability that the outcome will be a 6 or an odd number.
Now, let's consider the same event
Note that
There is a 5/6 probablity that the outcome will be in
In the previous examples, you noticed that for our dice example, it is easy to use these fairly straightforward probability formulas to calculate probabilities of certain outcomes.
However, if you think about our text message example, things are less straightforward, e.g.:
"What is the probability of sending less than 20 text messages in a day?"
This is where the probability concepts introduced here fall short. The probability of throwing any number between 1 and 6 with a die is always exactly
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 large numbers and how it can be used to calculate probabilities, along with the three probability axioms.