Joint Probabilities

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Do curso por Duke University

Data Science Math Skills

1685 classificações

Duke University

Data Science Math Skills

1685 classificações

Data science courses contain math—no avoiding that! This course is designed to teach learners the basic math you will need in order to be successful in almost any data science math course and was created for learners who have basic math skills but may not have taken algebra or pre-calculus. Data Science Math Skills introduces the core math that data science is built upon, with no extra complexity, introducing unfamiliar ideas and math symbols one-at-a-time. Learners who complete this course will master the vocabulary, notation, concepts, and algebra rules that all data scientists must know before moving on to more advanced material. Topics include: ~Set theory, including Venn diagrams ~Properties of the real number line ~Interval notation and algebra with inequalities ~Uses for summation and Sigma notation ~Math on the Cartesian (x,y) plane, slope and distance formulas ~Graphing and describing functions and their inverses on the x-y plane, ~The concept of instantaneous rate of change and tangent lines to a curve ~Exponents, logarithms, and the natural log function. ~Probability theory, including Bayes’ theorem. While this course is intended as a general introduction to the math skills needed for data science, it can be considered a prerequisite for learners interested in the course, "Mastering Data Analysis in Excel," which is part of the Excel to MySQL Data Science Specialization. Learners who master Data Science Math Skills will be fully prepared for success with the more advanced math concepts introduced in "Mastering Data Analysis in Excel." Good luck and we hope you enjoy the course!

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Introduction to Probability Theory

This module introduces the vocabulary and notation of probability theory – mathematics for the study of outcomes that are uncertain but have predictable rates of occurrence. We start with the basic definitions and rules of probability, including the probability of two or more events both occurring, the sum rule and the product rule, and then proceed to Bayes’ Theorem and how it is used in practical problems.

Probability Definitions and Notation7:39

Joint Probabilities6:24

Conheça os instrutores

Daniel Egger
Executive in Residence and Director, Center for Quantitative Modeling
Pratt School of Engineering, Duke University
Paul Bendich
Assistant research professor of Mathematics; Associate Director for Curricular Engagement at the Information Initiative at Duke
Mathematics

Next we're going to talk about joint probabilities.

Joint probabilities are the probabilities that two separate events

from two separate probability distributions are both true.

So, we're looking at the probability that A is true and

B is true and our notation allows us to abbreviate this

simply placing both capital letters together with a comma between them.

This is read as the joint probability of A and

B or the probability that A is true and B is true.

An important point here to note is that

the ordering does not matter in joint probabilities.

So also, I want to point out that our standard notation for probability

uses capital letters to refer to the entire probability distribution.

That's the entire collection of exclusive and exhaustive statements and

we use lower case letters to refer to the individual statements and

we can reference an individual probability the same way, okay?

So, what we've written here suggests

that the joint probabilities

are equal for regardless of which order we write the terms.

Or which order we consider the probability of distributions and

the same is true of each individual probability within the distribution.

So the probability of X1 and Y1 occurring together is the same as

the probability of Y1 and X1 occurring together.

This may perhaps seem obvious to you but, it turns out to be a very,

very useful principle as well.

Now, we're going to talk about the definition of the independence

of two probability distributions.

If two probability distributions are independent, knowing the outcome of

one does not our belief in the truth of the other.

Therefore, knowing the outcome of one,

does not change the probability of the other.

When two probabilities are independent,

you can think of them as being unrelated and unconnected.

I'm tossing a coin.

It has a 50 percent probability of coming up heads.

And I'm rolling a die and

it has a 1 in 6 probability of coming up as a 3 ,okay?

The joint probability heads and

3 is = to the probability of heads

which is 1/2 times the probability

of coming up three on a six-sided die

is 1/6 and that = 1/12.

So, for probability problems involving independent distributions, this is

the formula to calculate the probability that both events happen at the same time.

We have a special name for the product of the two probabilities

that is known as the product distribution, okay?

So, when the joint distribution equals the product distribution,

the two probability distributions are by definition independent.

We often use Venn diagrams.

These are diagrams that show the intersection and union of sets to

illustrate the intersection of two sets of events.

And, get developed some intuition about what it

means to say the probability that both events occur.

So for example, if we have the probability of flipping a coin and

it comes up heads and the probability of tossing a die and it comes up three.

Okay, we can represent this area as the product of the two, okay?

You can think of this outer area as the universe of all possible outcomes, okay?

And this gives us the probability that both

of the events that we have defined as relevant occur at the same time.

Perhaps the Venn Diagram is even more useful in capturing the concept of or

probability.

What is the probability that our coin comes up heads or,

now interestingly, this is written with a plus sign.

But in probability, what we're doing is we're looking

at the probability that either one event occurs or

the other occurs or both occur, okay?

So, we're interested in the probability that

the coin comes up heads or the die comes up 3, okay?

Well, if we simply added the two circles together,

we'd count the central area twice.

So to avoid that, we subtract the central area from the sum of the two circles.

So we are subtracting the joint probability which as we

know is equal to the product probability 1/12.

So, the probability that

either the coin comes up heads or

the dye comes up 3 would be

1/2 + 1/6- 1/12 or

6/12 + 2/12- 1/12 = 7/12.

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