Probability Theory 9 | Independence for Events

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This is my video series about Probability Theory. I hope that it will help everyone who wants to learn about it.
This video is about probability theory, also known as stochastics, stochastic processes or statistics. I keep the title in this general notion because I want cover a lot of topics with the upcoming videos.

Here we talk about the important concept of independence of events. We define it for two events but also for infinitely many.

00:00 Intro
00:19 Visualization (Independence for events)
03:48 Definition of independence
04:52: Example (discrete case)
07:38 Continuous case
10:52 Outro

#ProbabilityTheory
#Analysis
#Calculus
#Mathematics

I hope that this helps students, pupils and others. Have fun!

(This explanation fits to lectures for students in their first and second year of study: Mathematics for physicists, Mathematics for the natural science, Mathematics for engineers and so on)

[Hold_it]

Thank you for all the time you put into giving us such wonderful videos for free ❤️

[geoffrygifari]

Hmmm sometimes i see problems in probability where set notations can be described in human language. Can we use 'and' for intersection, 'or' for union? "Two events are independent when the probability of A *and* B equals the probability of A times probability of B" ?

Thanks for the video!

[Leslie.Green_CEng_MIEE]

I’m curious about the notation used at 3:21, consisting of an exclamation mark over the equals sign, but with no mention of its significance. I have seen (and used) a question mark over an equals sign to ask “Is it equal?” In this case we are looking for independence, which is assured if those (special) equalities are true.

This special equality is then used at 3:42 in the *“defining property for independence”.*

[sakcee]

I did not understand how at 3:00 the P(A|B) = 1/2, isnt the area of A inside B, 1/4 or 1/2 of 1/2?

[MrOvipare]

I like how the usual dart throwing game has been reduced to the simpler abstraction (but still a game) of throwing a point in a unit interval ahah :)

[deyomash]

I get the die example with two throws. But intuitively it's a bit strange. If i roll a 6 first, and ask the sum has to be 7. Then i behind my back FIX the outcome for the second roll, it HAS to be a 6. How is that independent of the first throw?

[geoffrygifari]

Honestly i did mistake being mutually exclusive for being independent before. Do you guys know how 'mutually exclusive ' is relevant in probability theory? What are some cases of independent events not mutually exclusive, and mutually exclusive but not independent?

[MrOvipare]

We often write down the probability space (Omega, A, P) and I watched the series on measure theory... but I'm still unsure of my intuition about A.
Let's say we are in the context of statistical mechanics describing a gas, could I say that :
1. Omega represent the set of the molecules considered in the problem
2. A represents the physical laws (dynamics, interactions) that govern the elements of the set. In physics term I would say it's the "partition function"
3. P is simply the measure, for example it could be the pressure, temperature, etc.
Does this seem reasonable?