Erika Duan 2022-04-19
We previously learnt that two events can have overlapping elements or be mutually exclusive (also known as being disjointed).
An example of two events being mutually exclusive is the observation that a coin can only land on either heads or tails when tossed. Mutual exclusivity therefore refers to the observation that only one or another distinct event is possible in a single outcome, which is different to the concept of independence.
When events are independent of each other, we refer to the observation that two or more events have occurred, where the occurrence of one event does not impact the occurrence of another event (also described as where the probability of one event occurring does not change the probability of another event occurring).
An example of independence is the real world knowledge that the outcome of two coin tosses are independent of each other. The observation that the first coin lands on heads does not increase the probability of the second coin also landing on heads, because each coin toss has a set probability of landing on either heads or tails.
The opposite of independent events is conditional probability. The
conditional probability of rain when the weather is cloudy or sunny is
denoted as
or
.
We instinctively understand that we are interested in predicting the
probability of rain when we have observed that it is either cloudy or
sunny and that these two probabilities may be very different
i.e.
.
This example illustrates the two properties of conditional probabilities:
- It describes the probability of event A occurring.
- It is known that event B has occurred and the occurrence of event B impacts the probability of event A occurring.
The source materials for this tutorial are:
- StackExchange answer on the difference between mutually exclusive and independent events
- Introductory Statistics online textbook by Saylor Academy
- Independence and conditional probability Youtube series from Khan academy
- The Probability for Data Science textbook by Stanley H Chan, specifically Chapter 2 on probability
- Introduction to probability theory GitHub resource by Michael Betancourt
- Introduction to probability theory Youtube series from MIT