Conditional Probability
Introduction to the joint and conditional probability. This lesson will teach you how to apply probability with events. We will also cover the Bayes Theorem.
We'll cover the following...
In this lesson, we go over an advanced concept of probability. First, let’s start by understanding joint and marginal probabilities.
Joint and marginal probability
Marginal probability is the probability of one value of a variable irrespective of other variables. It is the probability of the outcome of each individual variable.
Joint probability is the value of two or more variable values coming at the same time. It is the probability of the intersection of the outcome of variables.
Let’s understand these concepts with an example. Suppose we are collecting data for the car type of a driver and location in which he is driving his car. We have two types of cars:Hatchback and Sedan. Also, we are doing a study in 3 cities: New York, San Francisco, and Chicago. Each of the drivers has these two features: driving city and car type.
Consider the following contingency table:
Consider the following table:
Blue cells are located at the margins. They present row and column totals separately. They are called marginal values. So, marginal values represent the count of one variable in the dataset without considering other variables. For example, the total number of drivers in New York is 148, irrespective of what car they are driving.
We can divide each of these cells by the total number of drivers. Consider the table below. Here, the marginal probability of a hatchback car is 0.465.
The probability that a car is a hatchback and is being driven in New York is 0.2125. This is known as a joint probability. Joint probabilities are highlighted in the green cells.
The sum of all the joint probabilities is 1. Every joint probability is a disjoint event from other joint probabilities. The joint probability set is also an exhaustive set, as no other option is available. Hence, the sum of all joint probabilities is 1.
Interview question:
Interview question:
If you are given all joint probabilities, can you calculate marginal probabilities?
Quiz: Consider the following data of degrees
| Doctor | Engineer | Arts | |
|---|---|---|---|
| Class-A | 15 | 13 | 22 |
| Class-B | 26 | 45 | 36 |
| Class-C | 36 | 22 | 25 |
This represents data of 3 higher degree schools. We have considered only those students who are involved in a doctor, engineering, or arts degree.
What is the joint probability that a person belongs to class-A and is a doctor?
0.054
0.0625
0.15
0.1875
Conditional probability
Conditional probability is the probability of an event happening given that another event has already occurred. It is denoted by P(A|B). In other words, it is the probability of event A given that B occurred.
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