Are A and B Independent?
The Venn Diagram above shows the probability of Events A and B occurring.
Are they independent events?
If you said yes, you probably pictured independent events the way I first imagined them — as two events that never occurred simultaneously. However, this is not actually what independence of events means when dealing with probabilities.
To start with, let’s look at some definitions:
- Conditional Events: If you know the probability of one event happening, it tells you something about the likelihood of another event happening.
- Independent Events: Knowing the probability of one event happening tells you nothing about the likelihood of another event happening.
- Mutually Exclusive Events: If one event happens, it tells you that the other one did not occur.
However, it might be more helpful to look at some examples.
Independent Events
A restaurant collected data on the orders of its customers. Below are the probabilities of the customer being a student and of ordering either Tacos or Burgers.
In this scenario, Meal Choice and Student Status are independent events. This means if a customer is a student, it neither increases nor decreases the probability that they will order a specific meal.
- The probability of any customer ordering a Taco is 20/60 or about 33%
- The probability of a student ordering a Taco is 3/15, which is also about 33%.
Therefore, if a waitress sees a student, they will not be able to predict whether they will order Tacos or a Burger. The two events are independent.
The Venn Diagram of this situation looks something like this:
The fact that a customer is a student doesn’t affect whether they order tacos. However, there will be some students who order tacos. So, it actually makes sense that the events would overlap sometimes.
Conditional Events
They also collected data on payment methods. The table below shows the probabilities associated with student status and payment types.
The probability of any customer paying with a card is 43/60 or about 72%. However, the probability of a student paying with a card is 13/15 or about 86%. So, if a waitress serves a student, they could predict with a greater certainty that they will be paying with a card than if their customer is not a student.
Here is a rough idea of what a Venn Diagram showing this scenario might look like:
Mutually Exclusive Events
Now, let’s look at mutually exclusive events. In the table below, there are two examples of mutually exclusive events.
The first is whether someone is a student or not. Someone can either be a student or not. They can’t be both at the same time. The same is true for payment methods (at least in the context of this example). Someone could either pay with cash or a card, but not both.
A diagram of this situation would look something like this:
If you were surprised that the first diagram did not depict independent events, you might want to watch this series of videos from jbstatistics. He does a fantastic job of clearly explaining basic concepts related to statistics.
