The quick proof of Bayes' theorem

Including some added words on independence.
An equally valuable form of support is to simply share some of the videos.

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If you want to check it out, I feel compelled to warn you that it's not the most well-documented tool, and it has many other quirks you might expect in a library someone wrote with only their own use in mind.

Music by Vincent Rubinetti.
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If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people.

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Various social media stuffs:

[tardx]

When I fly, I always carry a bomb onto the plane because the chances of there being 2 bombs on the same plane is TINY

[karhukivi]

You might think that this is just some mathematical issue, but the point made at 1:52 can have dramatic consequences. An English solicitor Sally Clark was convicted of killing her two baby sons on this same statistical error given as "evidence" by an eminent doctor (not statistician) and jailed. She served three years of a longer sentence before the Royal College of Statisticians intervened to point out the error of multiplying the probability of two children dying of the same cause, which was sudden infant death syndrome, and more likely to be genetic and in both children. Her career destroyed, she died from alcohol intoxication (suicide?) and the "eminent" doctor was struck off the medical registration list.

[_kopcsi_]

ok so in short: simple multiplication of the probabilities is just a special case of Bayes’ theorem, when we have independence between events. therefore: P(A)=P(A|B).

[l.]

Can you maybe do a series about probability/statistics?

[Lightnx]

When I was first presented with this theorem, this is exactly the first thing I thought of. I thought "It's so obvious, how is this even helpful?". It was only later when I saw its many applications that I inderstood that a formula doesn't need to be complicated in order to be useful.

[nibblrrr]

1:34 *Poisson* distribution, *normal/Gaussian* distribution, *binomial* distribution, *expected value* is linear, *variance, * *Shannon entropy* _H, _ *Beta-binomial* distribution, *Benford's law, * *covariance* (with the definition of expected value already plugged in)

[DynestiGTI]

Eagerly awaiting part 2. This is easily my newest favorite series of yours.

[RagaarAshnod]

I've been trying to find more information about the proofs for simple statistics for ages! Like how do you come up with even the simple things like variance, and standard deviation. So I love that you discussed this! It's a good start!

[nathantang]

I love how you teach the intuition behind many topics that may be somewhat inaccessible at first. I'm curious, do you have any tips on how to develop these intuitions and what should you be looking out for in order to notice that observations that will propel your understanding deeper? What is the process of gaining intuition like?

[specific_pseudonym]

Hopefully just a bit of constructive feedback: The bit at the start moving bars around representing where A and B are in terms of their probability is...not exactly intuitive. I'm still not entirely sure what the goal was with it; was an overlap intended to be shown? If so, well, it's just hard to see an overlap in that type of diagram.

Good video nonetheless though! Thanks for it and especially the main one.

[Alysio]

Great videos (both of them, as well as all the previous ones actually). Because I just can't help it, I will point out that you missed a minus sign at 1:35 in your Gaussian distribution's exponential ;)
And even though I thought of this explanation during all of the main video ("yeah, it's just writing the definition of P(A ∩ B) in both possible ways"), I will say that I never interpreted Bayes' theorem as the way information gets updated given new evidence ; and I definitely hold the morale "Evidence should not determine beliefs but update them." very close to my heart.

Thank you for the great content and math teaching, and long live high-quality - high-creativity education such as the content you provide ! ~ :D

[ferax_aqua]

Two 3b1b in 10 minutes.
Is this early Christmas……

[raconteurhermit]

From bayes rule, Most important use is to calculate Probability of Cause given Effect
P(cause|effect) =
Since most of the scenarios in any experiment or simulation, its easy to get value of effect by applying cause , i.e P(effect|cause) can be easily calculated by actual experiment
Now using Bayes theorm one can calculate P(cause|effect)


An example
Question:You are working on your laptop, and you hear a whirring noise from it, whenever you boot it. You go to a technician and he tells you that 80% of the cases of laptop repair that come to him, have the same problem of whirring noise. You assume that the issue is from your hard-drive, and you read online that 20% of all laptop issues are hard-drive related, while the problem of getting a whirring noise from the laptop on boot from a hard-drive related issue is 50%. Given this information, you calculate the probability that you have a hard-drive related issue in your laptop ?
Answer:This is a simple application of bayes rule.
Here whirring noise is an “Effect” or symptom, and Hard-Drive related issues serve as a probable “Cause”. Then P(cause) = 20% and P(effect) = 80%, and P(effect|cause) = 50%. From bayes rule,
P(cause|effect) =
= 0.5*0.2/0.8 = ⅛

[technoultimategaming]

hello. I just wanted to say thank you. Using your "Essance of Calculus" helped me a lot in math. Especially the taylor series. It helped me understand eulers formula and that helped me with complex numbers today. I had to use De moivers theorem to solve roots. Instead I did it using eulers formula thanks to you

[Koisheep]

Personally I think no explanation of why it is true is better than explaining what a joint probability is. It just rolls out naturally that you may want to find an "event-swapping" law, since P(A|B) has two "variables" (A and B). Or maybe I'm just too deep into the math at this point.

[VincentZalzal]

I am glad you posted this footnote. I actually understand a lot better Bayes' Theorem through this Venn diagram and by thinking of a probability as a proportion between two areas of the Venn diagram. It's always good being able to explain some concept in more than one way.

[changyang]

The fact that P(B)*P(A|B)=P(A)*P(B|A) is in fact how I remember and write Bayes formula each and every time. I always start with this and transpose either P(A) or P(B) to the other side of equation to get the equation need.

[edwardlau]

I start formulating a geometrical question when I start visualizing Bayes' theorem.
With a square or circle of area 1, while the area represents probability.
Then we cut the square or circle with two straight lines to divide the shape into 4 areas: P(H && E), P(E && !H), P(H && !E), P(!E && !H).
Given their proportions, return the coordinates of the vertices of the cutting edges.
This might be a more visually fluent representation at the cost of intuitiveness.

[michaelantoun]

In the example given at 2:21, I noticed that the animation shows an increase in probability only horizontally, i.e. that your brother dying of heart disease means you are more likely to die of heart disease. Does it not also hold that if you die, of heart disease, your brother is then more likely to die of heart disease, implying a stretch not only horizontally but also vertically? If my thinking is incorrect, I'm very interested as to why.

[omargaber]

mathologer and 3Blue1Brown are my life.