3.09 Conditional probability

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From the course by University of Amsterdam

Basic Statistics

1704 ratings

University of Amsterdam

Basic Statistics

1704 ratings

Course 3 of 5 in the Specialization Methods and Statistics in Social Sciences

Understanding statistics is essential to understand research in the social and behavioral sciences. In this course you will learn the basics of statistics; not just how to calculate them, but also how to evaluate them. This course will also prepare you for the next course in the specialization - the course Inferential Statistics. In the first part of the course we will discuss methods of descriptive statistics. You will learn what cases and variables are and how you can compute measures of central tendency (mean, median and mode) and dispersion (standard deviation and variance). Next, we discuss how to assess relationships between variables, and we introduce the concepts correlation and regression. The second part of the course is concerned with the basics of probability: calculating probabilities, probability distributions and sampling distributions. You need to know about these things in order to understand how inferential statistics work. The third part of the course consists of an introduction to methods of inferential statistics - methods that help us decide whether the patterns we see in our data are strong enough to draw conclusions about the underlying population we are interested in. We will discuss confidence intervals and significance tests. You will not only learn about all these statistical concepts, you will also be trained to calculate and generate these statistics yourself using freely available statistical software.

From the lesson

Probability

This module introduces concepts from probability theory and the rules for calculating with probabilities. This is not only useful for answering various kinds of applied statistical questions but also to understand the statistical analyses that will be introduced in subsequent modules. We start by describing randomness, and explain how random events surround us. Next, we provide an intuitive definition of probability through an example and relate this to the concepts of events, sample space and random trials. A graphical tool to understand these concepts is introduced here as well, the tree-diagram.Thereafter a number of concepts from set theory are explained and related to probability calculations. Here the relation is made to tree-diagrams again, as well as contingency tables. We end with a lesson where conditional probabilities, independence and Bayes rule are explained. All in all, this is quite a theoretical module on a topic that is not always easy to grasp. That's why we have included as many intuitive examples as possible.

3.08 Joint and marginal probabilities6:02

3.09 Conditional probability4:49

3.10 Independence between random events5:13

3.11 More conditional probability, decision trees and Bayes' Law8:12

Meet the Instructors

Matthijs Rooduijn
Dr.
Department of Political Science
Emiel van Loon
Assistant Professor
Institute for Biodiversity and Ecosystem Dynamics

In this video, I'll explain the concept of conditional probability and

show how it can be calculated.

The term conditional means depending on something else and has more or

less the same meaning in the context of probability

as it has in everyday language.

Its formal definition is as follows.

The probability of an event given that another event occurs.

And this is its mathematical notation.

The probability of an event A given that B occurs or conditional on B.

The vertical line is shorthand for the word given, or conditional.

A conditional probability of A given B is calculated by the probability that

events A and B occur together, divided by the probability that event B occurs.

So the joint probability of A and B divided by the probability of B.

It can also be illustrated with this Venn diagram, which emphasizes that the joint

probability of A and B can only be smaller than or equal to the probability of B.

Let's apply the equation to a familiar example.

You've made account of different activities by the people on the beach, and

you also distinguished people by gender.

You have turned the results into a table with probabilities.

Here it is.

So with these variables, activity and

gender, what would be an example of a conditional probability?

It are those probabilities for

the case where you know the outcome of one variable, and

then want to calculate the probability of the other variable occurring.

Let's pick a concrete example.

You would estimate the probability that the person is resting if

you know that person is male.

To calculate it, you would apply the equation and

divide the joint probability of resting and male by the probability of male.

Hence, 0.3 divided by 0.45.

To calculate the conditional probabilities for the other activities, given

that the person is male, the equivalent calculation is made as shown here.

And for females, the conditional probabilities for

the other activities would be calculated by dividing with 0.55.

The resulting conditional probabilities for

the activities given gender are shown here.

Now I have a question for you.

Can you calculate the conditional probabilities of gender, given activity?

This is the original data which provides you with required information.

What you should do here is to consider a joint probability.

For example, the case of being male and resting.

And then divide this joint probability by the marginal probability for

the relevant activity.

In this case, resting.

Next, this should be done for the remaining five joint probabilities.

This gives you the following six conditional probabilities.

So based on joint and

marginal probabilities, you can calculate conditional probabilities.

But the equation for

conditional probability, simple as it is, has a little bit more to offer.

It is at the same time the general equation for

calculating joint probability for both independent and non-independent events.

Take a look at the equation again.

If you multiply both sides with the marginal probability for

event B, you have an expression to calculate joint probability.

It implies that if you are given a task to find a joint probability for events A and

B where you cannot assume independence between A and B, you would

need the probability for A as well as the conditional probability for A given B.

Let me summarize what I have explained in this video.

Conditional probability is the probability of an event

given that another event occurs.

While it may not seem that special on the face of it,

it is at the heart of many probability calculations.

Mathematically, the conditional probability of A given B

equals the joint probability of A and B divided by the probability of event B.

The definition for conditional probability is at the same time the general definition

for joint probability of both independent and dependent events.

The probability of A conditional on B can be considered

as the probability of A in the reduced sample space where B occurred.

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