Mutually exclusive events and independent events are not the same.

Mutually exclusive events and independent events are not the same, therefore don't mixed up [1]. Recently, I started to read a book about probability. In the second chapter of the book [2], I have already found my big misunderstood. I didn't know this for a long time. I am not good at probability. Maybe because I missed very fundamental things like this. I thought mutually exclusive and independent are the same thing. Here, I would like to explain they are completely different concepts.

Mutually exclusive events are never happens at once.

An example of mutually exclusive events are

head of first toss of a coin
tail of first toss of a coin

One coin toss, head and tail at once never happened.

But independent events can happens at once. The definition of independent events is

This may not be 0. An example of independent events are

head of first toss of a coin
head of second toss of a coin

These two events are independent.

But, I also find this definition is kind of difficult for me. The definition said if the and-probability is equal to the multiplication of each probability, they are independent. However, how do you know the probability in general? I think it is hard to know exact probability from an observation. Maybe it's just definition, but, independent events are not so clear for me by this definition.

References

[1] Hisao Tamaki, Introduction to statistics for information science, for application of simulation algorithms. ISBN-13: 978-4781910123 (Japanese)

[2] Malvin H. Kalos and Paula A. Whitlock, Monte Carlo Methods, Volume I: Basics, ISBN-13: 978-0471898399

Comments

Gauss's quote for positive, negative, and imaginary number

Recently I watched the following great videos about imaginary numbers by Welch Labs. https://youtu.be/T647CGsuOVU?list=PLiaHhY2iBX9g6KIvZ_703G3KJXapKkNaF I like this article about naming of math by Kalid Azad. https://betterexplained.com/articles/learning-tip-idea-name/ Both articles mentioned about Gauss, who suggested to use other names of positive, negative, and imaginary numbers. Gauss wrote these names are wrong and that is one of the reason people didn't get why negative times negative is positive, or, pure positive imaginary times pure positive imaginary is negative real number. I made a few videos about explaining why -1 * -1 = +1, too. Explanation: why -1 * -1 = +1 by pattern https://youtu.be/uD7JRdAzKP8 Explanation: why -1 * -1 = +1 by climbing a mountain https://youtu.be/uD7JRdAzKP8 But actually Gauss's insight is much powerful. The original is in the Gauß, Werke, Bd. 2, S. 178 . Hätte man +1, -1, √-1) nicht positiv, negative, imaginäre (oder gar um...

Tezuka Osamu's Black Jack, "Shrinking"

I like several novel authors. My first favorite author is probably Teduka, Osamu. I still love him. The list grows by adding Hoshi, Shinichi, Agatha Christie, Hermann Hesse, and so forth. My first favorite article of Tezuka was Atom as most of the (boy's) Tezuka fans did. But my favorite is Black Jack. I try to summarize one story, it is still quite vivid in my memory. I first read this story when I was 13 - 15 years old. I re-read it at least several times since Black Jack is composed of many short episodes. The title should be "ちぢむ (SHRINKING)" or it might be "縮む(Shrinking)". (It is not so convenient to translate this to English, since English does not have a system to say the exact same word in several ways. So I just simulate it with capital letters.) Black Jack is a genius surgeon, but he does not have the license. In short, his medical activity is illegal. His skill is top level in the world, but, the fee is also out-of-law expensive. In the story ...

Why A^{T}A is invertible? (2) Linear Algebra

Why A^{T}A has the inverse Let me explain why A^{T}A has the inverse, if the columns of A are independent. First, if a matrix is n by n, and all the columns are independent, then this is a square full rank matrix. Therefore, there is the inverse. So, the problem is when A is a m by n, rectangle matrix. Strang's explanation is based on null space. Null space and column space are the fundamental of the linear algebra. This explanation is simple and clear. However, when I was a University student, I did not recall the explanation of the null space in my linear algebra class. Maybe I was careless. I regret that... Explanation based on null space This explanation is based on Strang's book. Column space and null space are the main characters. Let's start with this explanation. Assume x where x is in the null space of A . The matrices ( A^{T} A ) and A share the null space as the following: This means, if x is in the null space of A , x is also in the n...

Lx=d: SundayResearch

Search This Blog