2.1.3.2.1 - Disjoint & Independent Events

Note that disjoint events and independent events are different. Events are considered disjoint if they never occur at the same time; these are also known as mutually exclusive events. Events are considered independent if they are unrelated.

Disjoint Events

Two events that do not occur at the same time. These are also known as mutually exclusive events

In the Venn diagram below event A and event B are disjoint events because the two do not overlap.

Mutually Exclusive
Venn diagram
A visual representation in which the sample space is depicted as a box and events are represented as circles within the sample space.
Independent Events
Unrelated events. The outcome of one event does not impact the outcome of the other event.

Example: Freshmen & Sophomores Section

Let's consider undergraduate class status. A student can be classified as a freshman, sophomore, junior, or senior.

Being a freshman and being a sophomore are disjoint events because an individual cannot be classified as both at the same time. 

Being a freshman is not independent of being a sophomore. If I know that an individual is a freshman then the probability that they are a sophomore is 0; knowing that the student was a freshman provided information that influenced my prediction of them being a sophomore. 

Example: Class Status & Gender Section

Assume that there is no relationship between gender and class status. This means that within each class (freshmen, sophomores, juniors, seniors) the proportion of students who are men is consistent. It also means that within each gender the proportion of students who are freshmen, sophomores, juniors, and seniors is consistent.

In this case, we could say that the events of being a man and being a senior are independent events. Knowing that a student is a man does not influence the likelihood of him being a senior. Knowing that a student is a senior does not change the likelihood of them being a man.

There are some men who are seniors, so these events are not disjoint.