# Making Probability Mathematical | Infinite Series

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# Making Probability Mathematical | Infinite Series

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100 thoughts on “Making Probability Mathematical | Infinite Series”

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Career In Forensic Science

NOOO! I just got a man bun!

Ew that Latin

What I like about probability is the feeling after missing two 98% shots in a row after which your XCOM squad dies miserably. Also I like some angry comments of players who complain about the random numbers generator. That’s XCOM, that’s the life in our universe.

I've actually used that same probability logic that Pascal used to show why you should believe in the Christian God.

"Saying something has zero probability does not imply it's impossible."

Mind. Blown.

(Oh, and you forgot "town square"!)

Practically when you hit anywhere on dart board, you are simultaneously hitting a Planck area of points. So the probability of hitting any given point must be 1/(No. of plank Areas on the dart board) which is non zero. Just saying…

OMG… are humans normally THAT white? never seen IRL

4:43 What about squares that are places in a city? Like a town's square or times square?

Kelsey, ,great episode, and thanks for pointing out the man-bun should be no more. Unless you're Toshiro Mifune!

I am really uncomfortable with the probability of hitting a point being 0. It seems like it should be an infinitesimally small value, not 0. Having a probability of zero DOES mean an event is impossible NOT improbable. If you disagree please do state the difference.

I concur with the claim that random variables are "coarse", though (9:55). With a little bit more work you could have defined a continuous bivariate distribution on the board that has the required measure and is as smooth as you like. Otherwise, great video! Keep up the great work!

The Riemann Hypothesis could be a great subject for here. Seen some great videos about it, think you can do it better ^^

As you mentioned that a countable set of Infinitesimal small points have 0 probability(7:42) well that somehow made sense but when I gave it a thought and took it to the extreme I ended up confused with this question

If we added Infinitely many infinitesimal small points all over the board, it would have 0 area thus 0 probability! But!

It is really obvious that we could hit these points 100% of the time since there are infinite of them Everywhere On the Board..

Thx for your amazing Channel ^_^

I think pseudo-random number generators are interesting too. Have people studied the probability of those generators actually producing "random" numbers?

(patterned numerically after Plato's allegory of the cave and quantum mechanics with everything in parenthesis not known to the "prisoners") We are in total darkness. (before us is a pair of dice in a bowl, on a table.) Only the dots on the top of the die illuminate – when they are at rest, we have no knowledge of the shape of the dice, if there exists anything between the bottom (of the table) which can only be hit crudely (by our palm). we hit the table, two numbers transform in orientation, position, and value. The table is hit many more successive times, the behavior of the "dots" is observed over time. How do we conjecture on the natures of the shapes of the dice, table and maybe (the bowl)? probabilities and statistics.

Please more videos like this!

Etceterais pronounced eht-SED-er-uhh, not ex-SET-er-uhh.Correct pronunciation helps maximize the professionalism of your videos.

Favorite? It has to be the Monty Hall problem! It even bugged Paul Erdos, who was incredulous until running a computer simulation…

can you do an episode on Mirzakhani? thanks. <3

Do you perhaps have another channel on which you post more stuff?

Your take on Gödel's incompleteness theorems would be interesting.

if a 0% probability doesn't guarantee that it won't happen, does a 100% probability not guarantee that it will happen?

ET cetera. not ex etera

Did anyone else notice that the dice @6:10 are flawed? Opposite sides are supposed to add to 7 (1-6, 2-5, 3-4) but they have 5 and 6 opposite each other.

Can you do a video on Minimum Description Length?

I've always thought probabillity isn't real. If we look at the darts example, we can (in theory) accurately calculate where a dart will hit if we take into consideration the relevent data (Player's strength, dart's direction and weight distribution, wind, coriolis effect, etc).

I do however find probability theory convenient for engineering.

Since this video was about probability, and you mentioned that we want to make decisions rationally, I suggest doing a deeper exploration into this topic in a future video, and especially address Bayes' Theorem and Bayesian Reasoning.

In particular, the host, Kelsey, may enjoy reading E.T. Jaynes' book "Probability Theory: The Logic of Science", which develops a framework for Bayesian reasoning from the ground up. I'm currently watching a video lecture series 'reading course' of Jaynes' book, which helps lower the slope of the learning curve of the book. Search YouTube for 'Aubrey Clayton' or 'Probability Theory: The Logic of Science' to check it out. BTW: Jaynes, following R.T. Cox, develops the laws of probability

withoutaxioms, instead relying on an informal list of 'desiderata' which he formalizes in a general way, showing that Probability Theory is theonlyvalid mathematical way to reason about 'plausibility'. This is an interesting alternative to Kolmogorov, and does not rely on Measure Theory.I'm genuinely interested in morality and patterns of ethics and the probability of acceleration I guess? (and music)

Quantum mechanics

What is the probability that Kelsey is wearing that wedding ring to diminish the probability of random internet math nerds proposing?

When you start talking about the dart board and hitting a single point on that dart board: you need to properly define that point. If you expressly mean hitting exactly a 0-dimensional point and no other points around it, then that is impossible…but it's also somewhat redundant to mention.

If however, you mean that each point is infinitesimal in size (and this is typically what people do mean, as they are dividing the board into an infinite number of sections), and that the dart head is of a comparable size (i.e. also infinitesimal as already stated in the video), then the probability of hitting any specific point becomes the proportion of that point to the overall size of the board which is infinitesimal and clearly non-zero. Applying a real number field to this type of maths where you either take limits or involve certain types of infinities/infinitesimals is erroneous. Instead you should be applying a number field that includes these things as numbers (e.g. the surreal number field), which would lead to much less confusion over such things as inaccurate approximations are no longer used.

You should make a video on Bayes' Theorem. It is crucial for the understanding of the scientific method.

Thank for all the good you do.

I would like to learn more about

1. linear algebra (matrix calculations, algorithms, egen values, egen vector and so on)

2. fft or discrete fft (way and how it works, how to preserve amplitudes and so on)

3. approximating curves with sums of sin functions (how to kompres images using sums of sin)

I know these are topics for engeners and I use some of them every day. But I only know them good enough to used dem. I want to have a better understanding of them. And I believe these are topics many people would benefit from know.

Have a nice day.

Isn't the circle formed by many points? If so the some of all the points of an area should vê equal to it's area, so the some of the probability of multiplus points of an area should be equal to the probability of the dard being in some point of an area.

Like, we can say that a patch of 6 cookies is made of 1 cookie+ 1 +1+1+1+1, the somatory of the parts is equal to the total

If I told my parents to pick one number each of the subset : natural numbers, what is the probability of them picking any number x? lim h -> infinity p(x) = 2/h, but surely there must be a mistake in our mathematics here, we are missing the actual operation that my parents will indeed pick a number.

Can you do an episode on prime gaps?

Probable, also means probe-able, testable, deterministic (which is generally taken as an antonym when meant totally), e.g. Can you probe a jar for cookies, depends on whether your hand fits its mouth, whether there are cookies in the jars, etc. This is mathematics: that statements are logical, and, arithmetical, and the consistency is left as homework…

understanding nntaleb's tweets and MOOCs

A next-door-neighbour subject is that of reasoning given only probabilities and events. That is, given that event A has happened, and a bunch of probabilities, what is the probability of event B?

If i'm waiting for somone to arrive for an appointment how long do i wait after the agreed time before i cut my losses?

why is the probability of hitting a point not "one over infinity"?

Imagine you choose a number from 0-1 and the number you choose is the probability of some number to be one suppose the first number is one what is the chance of the next one to be one

7:13 the point (1/2 , 1/2) does not lie on a circle with radius 1/sqrt(pi) so it actually is impossible.

Am I the only one who have a crush on her? Hi Kelsey! You are so awesome! 😉

I think the sort of numbers that engineers and physicists use, with a "margin of error", and "significant digits", imply a 1-dimensional interval, at least on the marginal part.

The way that mathematicians use transcendental numbers, like "Real" decimals, seems analogous to the physics of wave-particle duality. A photon travels as a (higher dimensional) wave but interacts as a particle (at a 0-D point). Similarly a transcendental decimal transcends 0-dimensions and like a wave it can potentially intercept a range of points. Sure it may collapse to a single point, but this is a bit like the wave function collapsing. It's very interesting the way both of these examples cross dimensions, so to speak. It definitely serves a particular purpose in this way.

"Infinitesimally small" doesn't mean "exactly a point". A point doesn't have an size, whereas an infinitesimal area does.

In the video after this, you mention probability paradoxes.

This is a compilation of my favorites: https://www.quora.com/What-are-the-most-interesting-or-popular-probability-puzzles-in-which-the-intuition-is-contrary-to-the-solution/answer/Alon-Amit?srid=uB1Fo

I don't understamd math at all but i'm trying to save something because final exams are this year

Can someone please suggest some nice books about probability and statistics for a physicist?. I tried with "Fundamentals of Statistical and Thermal Physics" (by Frederic Reif) but… I would like something more axiomatic, Reif just loves doing approximations :/

I can't get over the idea that the probability of hit a point is Zero!!

I think of probability as a mathematical limit. Like, as the number of attempts N approaches infinity, the ratio of the number of times an event happens to N approaches the probability of the event occurring per attempt.

Probably not mathematically correct, but that's just the way it seems to me.

Continuous probability is a mathematical fantasy- useful for handling extremely short portions of possibility but really everything has a measure. A bad blind throwing dart player will not (if he hits the board at all) hit every point will equal probability (not that it really is the point of the exercise) depending on whether he is left/right handed and positioned relative to the board will introduce a consider bias to the placement of the darts- if LH then hitting more on the right and if below more on the higher part.

Only 100% and 0% probability remains true after the event has occurred that means whatever number you calculate it either happens or not so it's 100 or 0. Now why is probability important ?

Pascal probably wasn't aware of the prisoners' dilemma in game theory when he thought about god.

The dice at 1:10 are totally wrong. Opposite sides always have to add up to 7 in proper dice.

Somewhat Ironic how they talk about probability on a channel called infinite series. After all probability and statistics don't really apply to infinite series

I completely missed this when it first appeared due to what I think was insanity. I'm not sure the insanity is over but I just wanted to say that I think there is a hidden secret in coin flipping. It might even be tied to universe being a simulation. But remember, I'm not sure the insanity is over.

I have just come across your videos, and find them very informative. i wonder if you have ever done a video on what is sometimes called "The Monty Hall Game Show problem" Thanks for the excellent video.

As a physicist, I'm all for Solovay Model. Down with axiom of election! Just joking, of course.

Pascal is wrong. There's an opportunity cost in believing in God. You lose your freedom to dictate your own value and life direction. You'll be literally enslaved by a doctrine that's likely just a myth. The time and dedication needed to practice the religion will be forever gone. I see ZERO return on investment in this particular reasoning.

probability of a probability is that its a probability

More episodes on probability plz!

The problem of this version of game theory they take out the skill and free will of the gamer.

It's weird that PBS has to put advertising here, it sucks. Damn republicans.

The fundamental principle of probability is that as time goes by the probability does not change. If I have spun 5 heads in a row, we all shout that the next spin is 50/50, year 7 shout out. So why is it that when one studies those who are banned from casinos the common thread is 'missing outcomes'.

OW! my brain….ok take a break come back OW! my friggen brain.

Pascal’s Wager missed the obvious that he was not wagering on the existence of God, but rather the claims of an earthly institution threatening him with eternal torture for failure to accept their claims to Divinity. His loss is the loss is the waste of a human lifetime lived in fear to anyone making a large enough overbet coupled with the sociopathy to carry out horrific acts of violence like the Inquisition, Salem Witch Trials, etc.

“…I have hardly met a mathematician capable of reason.” — Plato

Philosophically, Descartes was an idiot, too.

I may probably understand this in future 😛

Pascal's Wage is so childish – his God is childish. "Believe and I will reward you" – and what exactly is this 'believe in God' thing? It's just a feeling that "I hope this foolish God will reward me for pretending that I gave a shit about him , while in fact all I care about is to get my reward". How can all this greedy rationalization can have any relation at all with God ?Strange that he couldn't see how naive his notion of God was. It's an interesting example of how one's greed and fear can make him hallucinate and believe that he is in some way 'religious' . Nope, it's all just childish self-deception. I guess he failed to see it because his mind was conditioned in such a way that he thought 'believe+god=reward' is some kind of axiom, or a truth. In my opinion all these three concepts are juvenile: 1) believing in something you don't know, 2) god – I don't even know what this word means or could mean. From others descriptions he must be some kind of person or entity which rewards you infinitely if you bribe him 3)getting rewarded by someone you just believe(you don't actually know) exists.. and for what? for believing, hhahahh.

Anyway, I still love this guy Pascal… I have much to learn from him. Great video!

Hmm. I think his rationalization could be better worded this way: it's worth investing your time and energy in discovering and obtaining Freedom/Truth , because if there is such a thing, then becoming it is worth any sacrifice, and if there is no such thing, then all you do will go waste anyway, death will take everything and you'll turn into nothing, so why bother at all? :)) .

9:22 you just gave a depiction of how the cortex learns a skill (except it does have a lovely reversable characteristic too it)

This channel always motivates me to study more math, sometimes school cant keep me interested enough, although I love math.

"saying something has probability 0, does not mean it's impossible"… yes it does; that is exactly what it means.

Love how, at 4:13 neither of the pictures shows Algebra or Geometry (Geometry that deals with constructions, that is)

Bayes theorem

A paradox arises from requesting an infinitely small area from a granular universe.

I had no idea that probability has had so many rigors apply to it so relatively recently.

Phil Dunphy is not a square!

This video should be shown at the first class of every probability course …

Why exactly must the chance you hit (x, y) with a dart be zero? If you were to integrate all the probabilities over the entire dartboard you'd get a total chance of hitting the board equal to 0, which is weird.

well, the probability that she reads this comment on such an old video is zero

Hmm 🤔 Of the tens of thousands of gods created by man… On which one of those shall I place my wager? Please advise!

Wow, now I can understand more Monte Carlo.

Im in love… (withh all respect) of you as well of the content in this channel and every pbs channel! Cheers from Mexico!

Let's talk about focal points!

Sorry.

Wow. Intelligence is so sexy.

so "Pascal's Wager" is basically the "it'd cost you exactly $0.00 to be kind" meme

If there are an infinite amount of exact points on the board, why doesn't that mean the probability of hitting one is infinitesimally small but not 0? Cause with the other examples like 1/4 the area fits on the board 4 times, with 1/2 it fits twice, so I was like if it fits on the board infinite amount of times it'd be 1/infinity? I'm just a layman so forgive me if I'm being stupid

Bayes Theorem is my favourite.

be there or be square came about because if you're not there you're not "a round". also it rhymes

My favorite aspect is it's predictive power

I suppose probabilities is basic to quantum computations.

Beautiful women. Thank you for being explaining maths!

I (we) miss you

Unfortunately very very basic. It's good for kindergarden pupils, though. Thanks for wasting my time!

Craps is my favorite thing about probability!😎

Pascal's wager fails, because aren't the probabilities of a belief rewarding god existing and a belief punishing devil existing equal?

If you say that there might exist a devil who rewards belief, but I say that there might exist a god who punishes belief.

If you say god doesn't punish belief, then the following is to be concidered.

suppose there exists an entity.

that entity can either reward or punish belief. the probability of that entity punishing is equal to the probability of that entity rewarding, right?

if that entity rewards, there is a possibility of it being god.

if that entity punishes there is no possibility of that entity being god, if you defined god to be belief non-punishing.

Pascal's wager is a classic example of cherry picking. Ignoring the existence of an entity who always punishes belief in god regardless of whether it is god itself.

Well it's not rational to believe in God, since in most major religions, the greater sin than to not believe in God, is to believe in a false God, and since we can't know which God is the real God, its more rational to not believe at all in a God.

My favorite part of probability?

AdDiNG FrACtiOnS

I was looking for something with more history to be fair. Especially the axiomatic approach of Kolmogorov's vs the limit approach for probability, and early origins of what happened afterwards.

I'm interested in the use of elementary probability to aid in rational decision making.