20:52 Here is what the computer did! First you start out with the sum from k=0 to n of (1 + tr)^k * (1 - sr)^(n - k) * Binomial(n, k) * p^k * q^(n - k) and then you can combine some terms because a^k * b^k = (ab)^k, getting the sum from k=0 to n of (p + ptr)^k * (q - qsr)^(n - k) * Binomial(n, k) Then, due to the binomial theorem, this is simply equal to ( (p + ptr) + (q - qsr) )^n And with a little simplifying you finally get (1 + ptr - qsr)^n :D

@EpsilonDeltaMain

10 ай бұрын

Amazing work!! That was much simpler than I thought!

@tobiaspeelen4395

15 күн бұрын

Thanks man

@pedi-kun3978

10 күн бұрын

i don't understand but i have to learn it

It was really cool to see that there were four different types of averages.

@VeteranVandal

9 күн бұрын

He actually showed an infinite amount of averages if you watch again.

I am thoroughly impressed and grateful for the quality of your videos. Recently, I have been voraciously studying probability theory, statistics, and machine learning. Your videos explain things so well that I had to say something. By the way, I did not want to use KZread's 'Thanks' feature-- I would much have rather subscribed to a Patreon or donated to you directly, but this was the only option, as I looked for other ways to support you! Thank you so much for your videos. They have completely taken priority over some of my lecture notes, the next chapter of Murphy's PML book that I'm on, and several videos by other creators. A few key videos of yours are next on my list for rigorous study thanks to the incredible bandwidth of knowledge transfer you provide. I also love your presentation style, with occasional humor and very well-placed context for certain problem solving decisions.

@EpsilonDeltaMain

Ай бұрын

Thank you so much for your donations

@atheoristspointofview7059

13 күн бұрын

Why does this sound exactly like an ai generated response...🧐

@monkeybob

10 күн бұрын

​@@atheoristspointofview7059 Because, to increase the chance of quality output, AI is told to default to formal prose like you'd find in an article or a letter, and not how people talk or text. This person wrote this like a letter. But that isn't surprising because they apparently put a lot of thought and effort in before even writing the comment. So, to me, it makes sense they wouldn't skimp out on the effort of writing the comment either. And, same as making a handwritten thank-you note, the appearance of formality and effort is itself part of trying to communicate gratitude.

Great video! Thank you so much that was great at elucidating so many scary stats and probability concepts in one nice, clean mathematical umbrella! Did a better job in 30 minutes than half a semester of courses

lost af but watched the whole thing

If you have one bad bet, your leverage is gone and you're immediately in debt and in a bad place. On average you'd be fine, but if you bet *everything*...

@richardboland1935

3 күн бұрын

It's a first passage problem

Very nice video, thank you! I think the proof of equivalence with maximizing the geometric mean can be much simplified by expressing the random variable ln(G) as a function of the ransom variable k, the number of wins

This video gave me a lot to think about including the intuition of putting my money under my mattress.

@VeteranVandal

9 күн бұрын

Well, only the house wins in a casino, and in capitalism you only win by betting on average perpetual* growth. (Perpetual* meaning while capitalism lasts) The other ways to make reliable gains are to bet on crisis happening, but that needs a lot of data analysis. You can do better than your intuition tells you, but not by a lot. In fact the more you have the better you can hedge. Funny how that is. It's almost like if the system that favours those that have inherently less risk of losing is favoured by them.

I found a counterexample for the continuous case: let X be a standard normal distribution N(0,1) which has one mode at X = 0 then Y = ∛X is a transformed random variable, and the real cuberoot is monotonic, but the probability density function of Y is f(x) = (3/√(2π)) Exp[-x^6/2] x^2 which has 2 different modes

@EpsilonDeltaMain

8 ай бұрын

Beautiful job!!

So i really disliked statistics class. This video, while i still have some dislike towards statistics, I want to relearn all of it again. Seems that statistics, out of all the branch of mathematics, is the most applicable in day to day life. Math class should teach more of the different types of mean, and how they can be used to solve questions from elementary school level to industrial real world stuff. It's not like it's the best, just another way of thinking the same issue and i find it really intriuguing thanks for the video :)

i like your ring theory video waiting for zero divisors 2

Actually, this also explains a lot of reinforcement learning based algorithms and other branches .. nice video

Sir, these were the best 30 minutes I spent in KZread all of my life. Thank you so much!!

@mido0700

Күн бұрын

Fr

I wish I already understood everything. It'll take me a few years or more to understand all of it. Do you have courses?

There are too few views on this excellent video

Nice job, mate! How did you edited your beautiful videos? Lemme know that

@ladyravendale1

10 күн бұрын

For combining the video and audio? Not sure. For making the video? To me it looks like Manim. No link since it will probably get nuked, but it’s the first result when you google "Manim"

3:25 sudden ASMR

Can one deduce to always hedge as a solution?

Hey, I'm a little confused at 26:03 . The median was 4 and so everything above 4 is at least half. But why is everything below 4 and 3 at least half as well?

@HaramGuys

Ай бұрын

Everything down to 3, so 3 and above

consider holding your microphone at a moderately-constant distance away from you. There are weird times where your one part of a sentence sounds as if you were in a room, and the others sound good.

what happens if the "average" is in the beneficial range but one, or more, local mean is negative? To put it another way : imagine a scammer who runs a "clean" game until a big spender arrives then influences the odds to his own advantage ... but no so much as to reduce the overall average below the inflection point.

20:52. The easiest way to do it is evaluate it using binomial theorem: You just multiply - p^k multiply (1+tr)^k together to get (p + ptr)^k - q^(n-k) multiply (1-sr)^(n-k) together to get (q-qsr)^(n-k). Then you just have E (n over k) * (p + ptr)^k * (q-qsr)^(n-k) = (p + ptr + q-qsr)^n = (1 + ptr - qsr)^n

The word "dice" is plural. One may refer to two or more "dice", but not to one "dice". The singular of "dice" is "die".

@hoots187

29 күн бұрын

🎉

@silver_colonel

14 күн бұрын

🤓☝

@eak74

9 күн бұрын

Although you are correct historically, dice is accepted as singular and is very commonly used in academic literature. Language changes.

@ProfessorDBehrman

9 күн бұрын

@@eak74 Believe it or not, I am well aware of changes in the language. Although it may be a losing battle, I believe in arguing against such linguistic monstrosities as "very unique" , and "one dice". I have already accepted as fait accompli the singular "they" but I avoid it when possible.

@scarm_rune

8 күн бұрын

and this is why french is a bad influence. not because of this exact word, but because it proves that it opened the gates to words with the same mental value as a dried bog

soooo in conclusion…

Hey diddle diddle, The Median's the middle. You add then divide for the mean. The Mode's the one you see the most, And the Range is the difference between.

It's a shame youtube doesn't let me like this video more than once.

This was a long way to go about saying, as many papers have already written about, just use a fraction of kelly, 1/2 or 1/4 whatever youre comfortable with really...

3:01 If the expected value is positive why will I eventually lose everything? In other words Why does act of maximizing expected payout lead to long term ruins?

@andy02q

12 күн бұрын

Let's say we flip coins. Heads I get your bet. Tails I pay you twice your bet. A game you would want to play as often and with as much money as you want which is very bad for me. Now next we assume that I own unlimited money. In this case you might think that you want to bet all of your money every single time, but you would be wrong. Let's say you have 1k $ saved. You bet. You win. Now you have 3k $. Now you bet 3k $. This time you lose. That was bound to happen eventually. Just bet again - oh wait, you don't have anything left. You should have become a millionaire, instead you lost everything. But with your strategy to become a millionaire you would have had to win 6 times in a row, that's very unlikely. If you only bet half of what you have, then you get to play much more often and thus abuse the game much better and have a much better chance at becoming a millionaire. Okay, let's dial back: Same game, you own 1000$. This time you only bet 500$. You win. Now you have 2500$. You bet 1250$. Lose. No problem, you still have more than you started with and even if you lose the next two games you are still likely to become a millionaire. Now you want to bet 625$. I refuse. It's a terrible game for me. Darn, maybe you should have bet everything after all. One more thing: In the first scenario you could have gotten very lucky and become a millionaire. Let's change the rules 1 more time to make it more obvious: You tell me your strategy and then we run through 1000 bets before you get to change anything. If you bet everything every time, then you might get incredibly lucky to have won an equivalent of the whole universe in gold at one point and I'd be still standing there, smiling, knowing that your not even close to not losing everything to me (and most likely I won't have to reveal to you, that I do actually not own infinite money and in fact not even the equivalent of the universe in gold, although that's much less). But if you just calculate the expected value, then you might get the idea, that betting everything every time is a great strategy.

@jeanf6295

11 күн бұрын

The expected payout is skewed by super improbable but gigantic returns. Playing at maximum risk is like playing one game where you have 1/2^N chance of winning 3^N times your bet, and 1-1/2^N of losing your bet. The main issue is that the law of large numbers does not apply here : the variance of this game grows much faster than N, the expected value is thus meaningless.

@hetanshthakore5886

11 күн бұрын

@@jeanf6295 Yes I understand what you are saying and thanks for pointing that out. But over here in the given example, the probability of winning P(W) is 0.59 which is pretty high. So why is such a high risk-high reward strategy detrimental, even after such a high probability of winning statistically speaking? Obviously practically you won't risk all your money in a bet even if chance of winning is 99% because if you lose [However unlikely] you will need to live on the roads.

@GK-gc9cv

3 күн бұрын

Take a game where flip a coin heads you 10x your money tails you lose everything. It's always optimal to play one more time but playing this optimal strategy will lead to losing everything. The longer you play the expected value grows and eventually you lose it all

@GK-gc9cv

3 күн бұрын

​@@jeanf6295yeah it creates a paradox. Where you should in theory play forever to get an infinite amount but with a probability that approaches 0, and a probability that approaches 100% that you lose it all

let profits ride until they become losses

Conclusion?

6:28 r > 0 not 1. Any value over 0 is positive gain.

I actually used this method to win a local church lottery.

Interesting math. & Ka-ching, huh?

Brb gonna go make a fortune

18:25 If Y = 1/X, then when X is 0, Y should be undefined, not 0.

Hello, what i need to study to understand the math of this video,? I want to learn how to compute gambling edge.

@brendanh8193

Ай бұрын

Algebra, simple probably and sequences and series are essential. Understanding conditional probably, optimisation and sensitivity analysis will also help but aren't essential as he explains it quite well.

@Suav58

9 күн бұрын

A lot, but this is only statistics. There are assumptions behind statistics. One of them is of an assumption of a "stationary process". You won't find an "interesting" and "strict" one in your lifetime hence all the statistics has its time validity intervals. All the known stationary processes have already been exploited. These, who got there first turned the field into some oligopolies. The "interesting" ones are subject to a fierce competition and some occasional "upsetting of the chessboard", so to speak. In other words, the "interesting" statistics are valid within some time framework which might be as small as a dozen milliseconds... nanoseconds even (HFT). You need some good approximation of an underlying dynamical system to be able to make reasonable guesses about statistics and time intervals. The beauty of mathematics lies in the fact, that these, purely mathematical, considerations apply in a multitude of other, non-financial situations, say, biological evolution.

ok, so you want to maximize final profit, not average profit. Is this a .... "duh!"?

Sick video, let’s make some money boys

Why I feel that I am dumb after looking this video?

@theLowestPointInMyLife

Ай бұрын

because its maths masturbation

Now I can become a trillionaire

@0MVR_0

13 күн бұрын

commit to one trillion activities that can guarantee you one dollar

@dennisestenson7820

9 күн бұрын

​@@0MVR_0too bad it would take about 6500 lifetimes if you were paid $1 for every breath you take.

Why does the voice keep changing really interrupts the flow of the vid

3:00 I . . . don't understand? You win more than you lose, and when you do lose, you lose 75% less than you would win. So in what world do you lose all your money on average? probability is confusing

@HaramGuys

10 ай бұрын

you will eventually lose just once if you keep playing. If you all in, that guarantees that you lose everything that one time

@Asdayasman

10 ай бұрын

You start with X money, and bet an amount B, either winning 2B, or losing B, (100% if you win, 100% if you lose, double or nothing). If you choose your bet B to be X - all your money, you can either end up with 2X, or 0. From this point forwards, X is equal to either 2X, or 0. In the former case, you can read this comment from the top down again and keep going. In the latter case, you have no money, and winning double of nothing leaves you with still nothing. If you start a game with zero, you end the game with zero. It is an endpoint, a black hole from which you will never escape. If you played an infinite amount of games, you would end up with the exact expected value by definition. In an infinite amount of games, because there's always a way to reach zero, you will eventually hit it, and then play infinitely more, ending up with zero every time, bringing your average down to zero.

@aguyontheinternet8436

10 ай бұрын

@@Asdayasman but you don't lose all your money in his example, only 25% of it. And it isn't additive, so losing 4 times in a row would only lose ~68% of your money, while winning four times in a row, which is more likely, will give 1500% more of your money back. Although I am starting to understand, if there's a way to reach zero, you'll eventually find it given infinite time. But if money was allowed to be continuous, instead of in discrete $0.01, then it would be likely to blow up to infinity.

@Asdayasman

10 ай бұрын

@@aguyontheinternet8436 My bad, I misunderstood the example, but you've got the point for sure - if there's a way to reach zero, you're gonna.

@HaramGuys

10 ай бұрын

​@@aguyontheinternet8436leverage/margin means ur playing with amount of money more than your own capital. Taking 4× leverage means ur betting 4 times the amount of money in your portfolio

Gambling is illegal in some countries

great video, i really needed a refresher in stochastics, however i was confused by the comment about Karl Marx wanting equal wealth distribution, which is untrue

Didn't understand a thing that happened

Critique: switching microphones every sentence was very distracting.

@federikus2928

3 күн бұрын

Bro if you are distracted by a slight change in voice you are the problem!

Simulating complex probabilities on a computer is easier lol

The over recording sucks, it'd be nice if you just did the entire paragraph again.

3:40 math strikes again! - How should we abbreviate RRRRisk? hmm.. r. What a great choice! - What about Gain? ...n? g?.. t! Yes of course, Tgaint, perfect sense! - Ok now we are at Lose, this should be easy. S for loSé - But why? so that 5:04 everything fits neatly on the screen as pqrst! - That completely throws me off! I've no idea what each letter is. - It's ok we can spend some time to describe each and get familiar with them. - Wouldn't all this be avoided if we use meaningful variable names? *Insert Boardroom Suggestion meme where reason is thrown out the window* I'll just go check if 3b1b posted anything new...

8:16 tell me you never read kapital without telling me you are still to misunderstand the concept of rates in surplus-value. Marx did not care about perfect equality of distributions, the notion is just slander.

@0MVR_0

13 күн бұрын

if the person with five dollars were to continue receiving that amount throughout a lifetime, at a certain point, the difference in realized accumulation would be sufficient for that person to enact laws that diminish the wage of the other or play the market in such a way that the same result is achieved.

no i don't want to be the insurance company stealing money from people

Irrational emotions and drunk people populate casino games... gambling is losing math is not necessary to avoid gambling 😅

@angelmarauder5647

Күн бұрын

It's a pretty evil empire.

Okay nerd tell me how to get rich at the casino and investing

Gambler’s fallacy

Um, you didn't include the fact that any major gambling or investing system is going to be rigged which reduces your probability of winning to 0.

@radme

5 ай бұрын

Pretty sure that’s what he meant when he said “you’ll lose everything to the house” in the first minute of the video.

8:13 "imagine one person having three dollars, and another person having five dollars. If Carl Marx saw this, he would want to equally distribute wealth." Nah, it would depend on which one of the people was Marx. If he had five dollars he'd say nothing. If he had three dollars he'd be saying the guy with five should give him two. I will spit on Marx' grave the next time I'm in the capital.

@Tezla0

10 ай бұрын

This example isn't related to Marx at all

@0MVR_0

13 күн бұрын

nothing to do with Marx so you're busy spitting and the drivel lands on your shirt.

As a mathematician, I’ve never heard so much math with so little understanding behind it before 🤮

Needlessly long, low information density

TLDR ... if you bet everything on a get rich quick scheme, you are likely to lose everything.

This is why theoretical traders can’t make it in the real world. Theory and praxis-heaven and hell!