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