3.03 Sample space, event, probability of event and tree diagram

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Do curso por University of Amsterdam

Basic Statistics

2117 classificações

University of Amsterdam

Basic Statistics

2117 classificações

Curso 3 de 5 em Specialization Métodos e Estatística Aplicados às Ciências Sociais

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.

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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.

3.03 Sample space, event, probability of event and tree diagram5:37

3.04 Quantifying probabilities with tree diagram5:32

Conheça os instrutores

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

Beaches can be eventful places, especially when the weather is good,

there are some people around, and there are things to do and see.

This type of beach is the backdrop for this video, where I'm going to explain

some basic concepts that help to find probabilities and

also a visual aid to do so, a tree diagram.

It has been a warm afternoon, and you could use a refreshment.

Luckily, on your beach, there is a stand which sells these.

It is almost sold out with only one type of ice cream and

two bottles of soft drink left for sale.

A bit unfortunate, you also find three persons lined up in front of you, but

on the positive side, the stand owner has limited sales to one item per customer.

Since you are desperately longing for a cold drink, you start to wonder,

what are my chances?

You have no idea about choices that the other customers will make, so

these are random events for you.

The first customer might choose either soft drink or

an ice cream, and after this choice, the second has the same options.

Next is the third customer's turn.

If the previous customers have both bought soft drinks,

she hass only one option left.

In the other situations, she might still choose between ice cream and

soft drink, and finally it would be your turn.

Through this tree diagram,

you have ordered all the possible outcomes of this random trial.

There appear to be seven all possible combinations of soft drinks and

ice creams in a group of three, except for three soft drinks.

The name of this list with all possible outcomes of a random phenomenon is

the sample space.

And if a random trial has discrete outcomes, as in this case,

a convenient way to picture the sample space is by listing all

the possibilities via a tree diagram as we've done here.

In a tree diagram,

there are intermediate outcomes at each split as well as outcomes at the final branch.

Any outcome, or a combination of outcomes,

is called an event, so an event is a subset of the sample space.

In this particular case, you are, of course, interested in the events unlucky,

no soft drink left for you versus lucky, you are able to buy a soft drink.

Any random event has a probability associated with it.

Also, these two events that are,

in effect, a combination of three small events.

A way to quantify probabilities for these events is through an experiment.

This could consist of observing the ice cream and

soft drink sales over a sufficiently long period, but there's no time for that.

While the first customer is making up his mind, you would just like to have your

answer now. So you resort to the alternative strategy,

to specify plausible assumptions about outcomes in the sample space.

You're going to assume that both outcomes for

each elementary event are equally likely and assign a probability of 0.5 for

choosing either ice cream or soft drink by each customer.

In any case, when assigning these probabilities per event,

you can rely on the general probability rules.

The probability lies between 0 and 1, and the total probability for

all possible outcomes, for example,

at all the options at each node in a tree diagram, equals 1.

This will quickly give you the answer you're looking for.

The answer may help you to decide whether you'd be optimistic and

cheerfully wait your turn, or start considering other ways to buy a drink.

However, what's important to remember, even after events have unfolded

themselves, you'll never know whether your assessment of the probability was correct.

You'd have collected information over three trials from

the customers in front of you, not many, and

therefore, not leading to a very accurate probability estimate.

Let me summarize what I explained in this video.

A sample space is the collection of all possible outcomes of a random phenomenon,

and an event is a subset of the sample space.

It corresponds to a particular outcome of a random variable or

a group of possible outcomes.

Each event has a probability, and

to find such probabilities you can use a tree diagram.

In a tree diagram, you make the sample space and

the assumptions about various events explicit.

To quantify the probabilities for

each event in a tree diagram, you can conduct experiments.

In the absence of these, you can sometimes specify plausible assumptions about

outcomes in a sample space and estimate the probabilities based on reasoning.

In any case, the general probability rules also apply to tree diagrams.

The probability of any event lies between 0 and 1, and a total probability

of all possible outcomes at the node in a tree diagram equals 1.

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