Why “probability of 0” does not mean “impossible” | Probabilities of probabilities, part 2

An introduction to probability density functions
An equally valuable form of support is to simply share some of the videos.

Curious about measure theory? This does require some background in real analysis, but if you want to dig in, here is a textbook by the always-great Terence Tao.

Also, for the real analysis buffs among you, there was one statement I made in this video that is a rather nice puzzle. Namely, if the probabilities for each value in a given range (of the real number line) are all non-zero, no matter how small, their sum will be infinite. This isn't immediately obvious, given that you can have convergent sums of countable infinitely many values, but if you're up for it see if you can prove that the sum of any uncountable infinite collection of positive values must blow up to infinity.

Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld

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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.
Download the music on Bandcamp:

Stream the music on Spotify:

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:

[bluebrown]

If you're curious, I never ended up making the third part of this. Or rather, I made part of it and thought it wasn't very good. The plan is to put together something like a probability series this year, where the beta distribution will surely be one of the topics. Thank you for your patience!

[DaisyAjay]

"The probability of the dart hitting the board is 1". You obviously haven't seen me play darts.

[bluebrown]

I have to imagine it's frustrating to follow this channel. I believe this is the third video in a row (excluding those on epidemics) that I ended by saying something like "we'll look at Bayesian updating in a continuous context in the next part". But whenever I think hard about the setup/prerequisite section of that video there's always something interesting enough to pull out to stand as its own video; there are just so many interesting topics here! Thanks for your patience, and hopefully, everyone gets that the goal here is to just hit as many fundamental ideas in probability as is reasonable.

Also, in parallel with making these probability videos, I'll be trying a very different sort of experiment on the channel soon...stay tuned.

[BenedictGS]

9 month later still waiting for part 3, it is okay take your time.

[barney]

Crush: You have 0% chance of being with me!
Me: So you're telling me there's a chance?

[jacemandt]

One of my favorite math jokes, relevant here:

A mathematician is a little drunk, and nudges the guy next to him at the bar and says, "Hey, think of a number. Any number at all." The guy says, "*Any* number?" "Yeah, any number."

"Okay, I got one, " the guy says. "Is it rational?" the mathematician asks?

"Ummmm...yes..."

"HOW UNLIKELY!!!"

[fukinyouup]

4:00

"Wait, it's all calculus?"

"Always has been"

[minerharry]

1:37 Those are some nice decimal places you have there. I recognized pi, obviously, followed by e and then later phi; but that third one was strange. 4.6692? What kind of a number was that? That’d have to be the square root of like 19, which is a weird number. Curious, I looked it up, and - with no context - the Wikipedia page for the Feigenbaum constants came up. Wikipedia pages on higher math are completely unreadable, of course, so I looked it up on YouTube and found a Numberphile video on it, because Numberphile has a video on every single number, and - because of a tiny little Easter egg in a video that I was rewatching for the second time - accidentally learned about a completely unrelated branch of mathematics and an incredibly strange phenomenon that arose therein.

I love the internet, and I love your videos

[adrift]

I like how whenever he says something, the little student pi's go like:

[shayanpoordian]

When I was trying to learn linear algebra, you put out a series solving all my confusion. then when I got interested in neural networks you put out a series which made me dive deeper and end up trying to learn stats. then you put a series on stats.

[saurabhmehta]

This series really makes probability and its probabilities click for me. Hopefully the long awaited part 3 will be uploaded soon :)

[inordirections]

Mr. BlueBlueBlueBrown,

Part three?

Sincerely,
Probability Stans Worldwide

[nanigopalsaha]

1:31 If the numbers after 7 seem familiar, they are:
0
1
π
e
The Feigenbaum Constant
φ

[dysxleia]

Whenever he has some arbitrary, long decimal number, he sneaks in π and e and golden ratio digits. 1:37 for example

[valeriodilecce]

I've been following this channel enthusiastically for years, yet I just noticed today in 2020 that the students/teacher pi creatures (2:37) are 3 blue and 1 brown. Yep.

[absolutelyproprietary]

Now achieving impossible has a whole new meaning

[frankbucciantini]

People: "What are the odds..."
Grant: "We gotta take a look at the probability density function".

[shakofarhad]

Just a former maths teacher talking into the internet void about probability:

To me it makes sense that the dart has a probability of 0 of hitting a specific point on the dart board. If you are aiming at a specific point, it means that you are betting on the fact that your accuracy will be on the level of atoms, and even smaller (because math has no Planck length). You literally are boasting infinite accuracy, which is impossible. That is why your probability of hitting that specific point is 0. But if you say "I am going to hit Bullseye". Then things change, now you are being reasonable. The bet is no longer on hitting the infinitely small point, but rather hitting an area which contains infinitely many of these infinitely small points. In some sense you have infinitely higher probability now since you have infinitely many small points. But of course in our real world we have the Planck length which means that we are never really talking about infinity, just very big or very small numbers. That also means that the probability is never truly 0, however it is extremely tiny. ^^

[SomeFreakingCactus]

Hey y’all, just thought I’d drop by for all the teens watching this video to say that you’re doing good. Keep it up. I’m learning this stuff at uni.

[irvinep]

I struggled for literally 5+ years to understand the shift from PMF to PDF and you just explained it in 10 minutes. THank you so much mate