The Law of Large Numbers and the Central Limit Theorem. Probability explained with easy to understand 3D animations. Correction: Statement at 13:00 should say "very close" to 50%.
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If we flip the coin an infinite times,the % of heads will be exactly 50% because no matter how lucky you are you can't get something more/less then that. Doing your rectangle analogy, there is only one way to order it,but while it goes to infinity, it always increases?
+Ibrahim Chahrour, thanks for the compliment. I am glad you liked my video, and yes I hope this will help clear up for people why large scale systems seem to be deterministic, even though quantum systems are not.
I recently created a Patreon account for people who want to help support my channel. The link is on my RU-vid home page. Also, in case, you have not already seen them, I uploaded several other videos recently. As always, for each video that you like, you can help more people find it in their RU-vid search engine by clicking the like button, and writing a comment. Lots more videos are coming very soon. Thanks.
огромнейшее спасибо за Ваш труд, Евгений! Я не представляю большего облегчения как после просмотра Ваших уроков. Вы досконально знаете физику и способны объяснить ее сложнейшие законы даже кошке.
I love your videos. I started watching them to help me with upgrading my science courses. Now that I am done with that, I just watch them because you make it all quite fascinating.
I came across this a few years ago when I made an application that generates sine waves radiating from any point in a rectangle (PictureBox) using a polar coordinate system. After I messed around with it by clicking all over the place at random, I ended up with a rather odd yet nice looking image which I then dropped into a histogram analyzer just for curiosity, to my surprise, the histogram resembled a nice bell curve. Well it looked sort of spiky, but it was nearly a bell curve, which just as explained on the video, the more samples, the smoother the curve became.
David Flores i made a program just for fun which pseudorandomly places a dot inside a circle and if one place is hit twice then point become darker and after lots and lots of dots every point in circle was hit and it was the same color
I think what's being said is like this: if an event is random, let's say raindrops falling on your rectangular flat roof in a steady gentle rain that lasts for hours, then to measure total rainfall you need not collect and measure all the rain (water) that fell on the roof, but only collect and measure at the center, then multiply by the area. If your roof was triangular then you could still measure in one spot and then multiply by area but the one spot would be offset from the center. The accuracy of the exptrapolation from measurement at one spot to predict total rainfall would increase with increased time of collection. With infinite time to collect the prediction would be infintely accurate....assuming the events were truly random. A more wide, rather than narrow bell curve would suggest uncertainty as to whether the sample was all that accurate. Also note: if you collected the rain long enough to get a very narrow bell curve, and you also collected and measured the actual rainfall from the correct spot (center for rectangular roof) , and the results differed significantly, then you could assume the rainfall (or whatever) was not random.
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The distribution of 1 2 1 for the number of heads after two flips seemed kind of familiar. Then with the distribution of 1 3 3 1, I realized something! An insight that struck me like lightning! The number of outcomes for the number of heads follows Pascal's triangle. For one flip, 1 outcome with no heads, 1 outcome with one heads. For two flips, 1 outcome with no heads, 2 for one heads, and 1 for two heads. For three flips, 1 outcome with no heads, 3 for two, 3 for three, 1 for four. For four flips, 1 for no heads, 4 for two, 6 for three, 4 for four, 1 for five. 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 ...
It is because the rows on pascal triangle are directly related to binary decisions such as coin flips. Say you are flipping a coin N times and you would like to know how many possible ways there are of flipping a total of X number of heads (or tails; it doesn't matter but you gotta choose one). You start by moving to the (N+1)'th row of pascals triangle, and look at each of the terms.The X+1 'th term in that row will be your answer. For instance the 1 on that top of the pyramid corresponds to the fact that if you flip a coin 0 times, there is 1 way to get 0 heads. the second row (1 , 1) correspond to the fact that if you flip a coin one time, there is one way to get zero heads, and one way to get one head. And for the third row (1,2,1), if you flip a coin 2 times, there is one way to roll 0 heads, 2 ways to roll one head, and 1 way to roll 2 heads. and so on for the rest of the triangle.
This is a great video that helps many people understand. Surprisingly and unfortunately, most physicists believe that when two essentially indistinguishable coins are tossed, the probabilities of getting ① both heads ② one heads and one tails ③ both tails are all 1/3. Look at the following question. [Question] Find the probability that the following three events will occur when two dice are tossed. ① both will be even numbers ② one will be even and the other odd ③ both will be odd numbers [The dice can be distinguished as A and B] The events are represented as (A number, B number). There are four types of events when only even and odd are considered. (Odd, Odd) (Odd, Even) (Even, Odd) (Even, Even) If each is considered to be equally likely, the probability of each occurring is 1/4. So the probabilities are ① 1/4 ② 1/2 ③ 1/4 (Result 1) Next, considering numbers 1 to 6, there are the following 36 types of events. (1,1), (1,3),(1,5) (1,2),(1,4),(1,6) (3,1), (3,3),(3,5) (3,2),(3,4),(3,6) (5,1), (5,3),(5,5) (5,2),(5,4),(5,6) (2,1), (2,3),(2,5) (2,2),(2,4),(2,6) (4,1), (4,3),(4,5) (4,2),(4,4),(4,6) (6,1), (6,3),(6,5) (6,2),(6,4),(6,6) If we consider each to be equally likely, then the probability of each occurring is 1/36. Therefore, the probabilities are ①9×(1/36)=1/4②18×(1/36)=1/2③9×(1/36)=1/4 (Result 2) (Result 2) is the same as (Result 1). [The two dice cannot be distinguished] (Odd, Even) is the same event as (Even, Odd), so there are the following three events when only even and odd numbers are considered. (Odd, Odd) (Even, Odd) (Even, Even) If each is considered equally likely, the probability of each occurring is 1/3. So the probabilities are ①1/3②1/3③1/3 (Result 3) For example, (1,3) is the same event as (3,1), so next, considering numbers 1 to 6, there are the following 21 events. (1,1) (3,1), (3,3) (5,1), (5,3),(5,5) (2,1), (2,3),(2,5) (2,2) (4,1), (4,3),(4,5) (4,2),(4,4) (6,1), (6,3),(6,5) (6,2),(6,4),(6,6) If we consider each to be equally likely, the probability of each occurring is 1/21. Then the probabilities are ①6×(1/21)=2/7②9×(1/21)=3/7③6×(1/21)=2/7 (Result 4) (Result 4) contradicts (Result 3).
Your videos are incredibly helpful. I am amazed with you ability to explain things and very thankful for you to share your gift with us. You make the more complex concepts very easy to understand. Thank you!
Thank you! As usual, very clear. I also like that the narrator speaks slowly enough to follow! Rather than speaking faster and trying to cram more into the same time. More may then be said, but less understood - by me, anyhow.
The balance in the universe makes me crazy. God puts the bell curve at the every law of the universe and this is all the reason why we are still exist. Somehow all the extremism are balanced with this way. Just amazing. Thank you for these videos.
Love from 🇮🇳India... Your content are amazing sir.I am really thankful to you for Presenting such theorems in 3D form. Please continue make such content i am in love with your content... ☺
Thank you very much Eugene. Although during my studies the bell curve and the theory behind it popped up many times, it was never so clear to me than it is now. Thank you.
I think I have understood from many quantum interactions, Newton's physics emerges because the probabilities of a body or an event with human perceptible dimensions become statistically almost if not completely deterministic, sorry for my English , is it rigth my think ? anyway this video is vero good 👍👍👍
@@ucondrew thanks but my opinion is rigth or not scientifically? someone thar graduate in epistemology of science or scientist will know it I hope ok happy new year
I wait eagerly for your videos. They are beautiful and intuitive! I never got such level of intuition reading any book or at College. Also love the animation and music. You have increased my interest in mathematics and physics by many folds. God bless humans with such good teachers. Thanks a lot!😊💯 Also would love to know about astrophysical concepts like black holes.
This summed it all ---> 21:06 "What we are typically able to observe in the universe around us is the average of the behavior of a very large number of subatomic particles. Although each of these subatomic particles is governed by the probabilistic laws of quantum mechanics, their average behavior becoms more and more predictable as the numbers of particles increases. This is why much of the universe that we are able to observe can be predicted through the deterministic laws of classical physics even though the underlying physics of subatomic particles is described by the probabilistic law of quantum mechanics."
Big take away, even though the underlying universe is random, aggregating many random particles when we know the probability distribution (and therefore what to expect) will be very likely to match our expectation.
Eugene you create good videos.Your videos helps layman understand concepts easily.thank you for quenching my curiosity by creating such simple but strong videos. I wish I could have access to videos at my schools days.
The central limit theorem is strictly speaking valid for processes with PDFs having a finite variance. Otherwise we end up with Levy walks and anomalous diffusion ... would be great if you could take a look into these as well. Great video nevertheless!
+Mateusz K. Thanks for the compliment. I am glad I helped with the understanding of quantum mechanics and relativity. Lots more videos are on their way.
+maxtomious, thanks for the compliment on the video. I have to keep the pace slow for people who do not have a strong background in mathematics or science. I especially try to keep the pace slow for certain critical concepts that are the foundation for understanding everything else that follows.
The pace is not only good for ones that don't have strong background in mathematics, but is also soooo soothing. Thank you Eugene! Your videos are very enjoyable and relaxing.
I can see the similarity to Quantum mechanics. I'll try to get my Ducks in a Row ! Hahaha You see this is why I need a Robot.Thank you I'm beyond my years!you have Increased my chances.
wow I am mindblown! Incredible video! Thank you very much for this :) But one question: Am I right to say that in the subatomic realms the particles behave in a strange "wave-ish" way and if we look at bigger things they behave very much predictable because there are more particles in a bunch, i.e. samples that constitute a bell shaped distribution? So now I wonder what is the transition between being in the unpredictable quantum mechanics world and the more "predictable" world that is governed by the laws of relativity? It is not a binary thing, is it? Either predictable or not? Either this or that? So there musst be a middle thing inbetween these realms, right? I hope this question is understandable :)
+Maja Muster, thanks for the compliment about the video. In reply to your question, there is no clear demarcation line. It is just that the more and more particles we have, the more and more unlikely it is that the observed behavior will deviate significantly from what is predicted. However, no matter how many particles we have, we can never eliminate this possibility to completely zero.
You have no clue how much potential this channel has, the problem is that rarely anyone hears of it. Try spending more time on getting your name known, your videos deserve MUCH more views and appreciation!
+Lowinator, thanks for the compliment, but I really don't know how to get more people to hear about my channel. If you have any ideas, please let me know. Right now, I pretty much rely on my viewers to share my videos and to help spread the word about my channel. Thanks.
who else watched this at 1.25x speed? my basic question is this - how can one even apply theory at the level of sub atomic particles in context of larger more macro spaces? no matter what math you use for empirical evidence ( probability being another theory) philosophically, what is the epistemological basis of quantum 'reality'? if we keep 'empirical proof' away from this, as a thought experiment.
Great video. Thanks. 12:51 It is not true that the more times we flip the coin the more likely it is for us to get exactly 50% heads. When we pass from two to three times the probability that we get exactly 50% heads decreases from 1/2 to 0. In fact, the probability of getting exactly n/2 heads in n tosses tends to 0 as n tends to infinity, for it equals (n choose n/2) over 2^n, which tends to 0 as n tends to infinity. It is one thing that relative frequency tends to probability with probability 1 as the number of experiments tends to infinity, which is a consequence of the strong law of large numbers, and quite another that the probability to get relative frequency exactly the same as the probability increases with the number of experiments, which is not true.
See Nick Kravitz' answer here: www.quora.com/If-a-fair-coin-is-flipped-an-infinite-amount-of-times-will-it-absolutely-land-heads-exactly-50-of-the-time-Is-it-possible-for-it-to-still-be-50-if-that-particular-infinity-is-odd
Fungo4 This is hardly posible because standard deviation is a weighted average of squared deviations. The point is rather that relative frequency approaches expected value with prob 1 but its probability to be exactly the expected value tends to 0.
There can be made a lot more different combinations of numbers that are close to the middle value. Thats why it peaks! awesome visuals and explanation "What we are typically able to observe in the Universe around us is the average of the behaviour of a very large number of subatomic particles." Nicely said. I wonder however, is all of it Bohr's ? Einstein thought differently of this, right?
There is a confusion between number of coins and number of times. We are flipping more coins. That is how number of particles in quantum studies matches.
one good thing-unlike other videos of yours, the breaks between the narrations were less. i would like you to elaborate a bit more on the probability density. still i loved it. but probability isn't the last thing in QM... i think we haven't actually understood it yet. full marks to the video.
i need the answer for this question below. Car color preferences change over the years and according to the particular model that the customer selects. in a recent year, 15% of all luxury cars sold were red. if 50 cars of that year and type are randomly selected, find the following probabilities: 1.At least five cars are red 2.At most six cars are red 3.more than four cars are red 4.Exactly four cars are red 5.Between three and five cars (inclusive) are red
Correction: At around 20:30 it says that the probability that the average will be exactly the at center of the bell curve approaches 100% as the number of samples increases, but this is incorrect. The probability that the sample average is *exactly* the predicted average is 0%, no matter how many samples you take. The correct statement would be the the probability that the sample average is within any given distance of the predicted average approaches 100%.
I understand everything in this video except for these: -@ 1:15s; How can an event with 0% probability still occur? 100% probability not necessarily occur? -@ 2:00s; Why is the probability of Pi, being picked on an infinite number-line, zero?
Your videos with caring emphasis on a different perspective on the mathematics is refreshing to see ! Keep them coming please ! P.S Could we see an intuitive video about complex numbers :)
First; thanks for another awesome video, Eugene! I was just wondering, what's the maths behind calculating the average of a triangular probability distribution (starting at 18:00), witch in this case was 6 and 2/3s? If you (or anyone else) don't have the time or space (no pun intended) to explain it, please tell me where I can learn about it.
@@aabishkararyal5846 Thanks! Since I posted the comment, I've started and completed 5 years of mathematics at uni. But I appreciate your time nonetheless.
Thanks for the beautiful demo. I always think that probability is only a practical concept when determining a complex event which we do not know all the factors influencing it. This video makes me wonder maybe probability is the reality, and certainty is only an illusion created by the aggregation of probability.
I did that once to determine whether the random number generator of my computer was truly random (although I think this may not be sufficient for that): I wrote a small script that added 1000 random numbers together and then divided the sum by 1000. Since the random number generator (I think I used /dev/urandom for it) generates numbers between 0 and 1, the outcome was pretty close to 0.50000
+Seegal Galguntijak, that proves that the average was 0.5, but that does not necessarily prove that the numbers were random. For example, we would get the same result if 100% of the numbers that the computer picked were exactly 0.5.
Martijn Bouman Thanks for throwing me the bone. But when I did that was in 2008 or so, I'm not really into the subject any more, so I'll just accept that /dev/urandom is possibly a pseudo random generator. Not a big deal (at least for my current usage).
So randomness means more certainty when looking at large sample averages. When something is not random it means more uncertainty. Just the opposite of what we think intuitively.
1:53 The number being picked being pi is quite literally impossible. Not 0%. I'm being nit picky, but what you said is the equivalent of flipping an infinite sided coin and it landing on its side, even with physics (momentum) being included. Pi is a function in a similar sense that 1/3 is a function, in that they cannot be shown in decimal form (1/3 is a function because it is showing 1 divided by 3, IE: .3333333333...); Pi, at least as we know it, goes on infinitely (an irrational number). I can drop that disagreement, but it would take sufficient argument.
Does this mean that the universe's end result is deterministic and there's no free will with that. No matter how the individual interactions between the constituent particles change, the result always tends to weigh in closest towards that one centred outcome on the Bell curve?
All the music in this video is from the free RU-vid Audio Library. I am not sure which song you are referring to, but it might be "Stale Mate." Most of the other songs can be found in the classical music section of their library. Thanks for the compliment.
Евгений, не знаю ответите на комментарий или нет, но все же спрошу, из видео совершенно непонятно, каким образом из равной вероятности выпадения чисел от 0 до 10 получается колокообразная кривая в районе пяти? Ведь если эти числа имеют равновероятный шанс выпадения при покидывании монеты с 10 странами (ведь чисел 10), то банальная перемешивание этих цифр вдоль прямой (например 735908214) где, например ноль мы можем разместить в середине этого ряда, покажет Колоколообразную кривую с пиком на этом нуле? Каким образом возникает этот колокол при том что каждое число имеет равную вероятность выпадения?
A question: re the random pick between 1 and 10. How is it possible to "pick" an irrational? You could never complete the selection. Interesting video Eugene and Kira.
While the chance of a heads or tails is approximately binomial wouldn't it be cool if you could measure the throw of a dice to predict the motion/speed of the particle itself? would you not able to predict a heads or tails then? Everything we see in nature follows a very distinct pattern...
SHOOT FROM THE HIP: - I just had the most profound thought. Pure abstract vector spaces cannot have a probability measure assigned to them ? Well sort of, only dynamical systems that approximate a vector space in which you can reasonably guess the restrictions that give rise to a non classical probability distribution function.
The music at 5:13 Suppose there are two cowboys, they draw their guns at each other and fire! What is the probability that at least one of them will die?
¿Cuál es la probabilidad de que al lanzar la moneda no salga ni cara ni sello sino que quede jurando o en equilibrio? Yo observé ese evento hace 25 años.
PLEASE make an episode about RANDOMNESS vs. PREDICTABILITY. Somehow these terms don't seem to be well defined. For example, radioactive particle decay (although unpredictable) can't be "RANDOM" since the half-life period is an exact CONSTANT for every element. So how on earth can we argue that a particle's decay is RANDOM? If it would be TRULY random (i.e. not dependent of/determined by ANYTHING, including particle properties like the element's mass etc.), any half-life time period would not EXIST at ALL. Don't we have to say that TRUE RANDOMNESS doesn't exist anywhere?
2:20 Величина, стремящаяся к нулю, и величина, равная нулю, - это не одно и тоже. Они не равны. В данном случая вероятность выбора числа пи СТРЕМИТСЯ, к нулю, тогда как вероятность выбора любого другого числа СТРЕМИТСЯ к бесконечности, потому что количество чисел в заданном интервале СТРЕМИТСЯ к бесконечности. Бесконечность - это не число, поэтому величина не может ей быть равна.
All the music in this video is from the free RU-vid audio library, and the names of the songs are the following. Road_to_Moscow Pachabelly Horses_to_Water Stale Mate
Nice explanation of classical probability, which is presented here as a fixed law that actually exists independently of the individual analyst. No update required. This follows from the assumption that is stated after about 0:45 seconds: "Suppose we repeat the same experiment...an infinite number of times...". Think about this for a moment. Can you really repeat an experiment exactly? Can you repeat a throw of a coin an infinite number of times in the same way? The classical theory of probability clearly assumes that the data is random and the result of a FIXED and DEFINED population distribution. In this way, procedures are devised that will in future have particular average characteristics in the event of repeated sampling. It is consequently not investigated whether the procedures allow valid processing after data processing. In summary, these fundamental assumptions have far-reaching consequences for our ability to understand mother nature. Since parameters only exist because we have invented a model, we should be very suspicious!
Glad to Be Normal there's no real line between them. As things get smaller, classical laws are increasingly likely to fail, so choosing one set of laws over another just comes down to hedging a bet. I am not a researcher, so I can't give you an absolute answer, but as far as I know, molecular chemistry is where most people switch to quantum laws. The ideal place to switch over would be at the size where quantum and classical laws have an equal likelihood of giving the correct answers. Keep in mind,though, that physics isn't usually used between small macroscopic objects (e.g. balls, bullets, pendulums) and quantum objects (e.g. atoms, nuclei, elementary particles) - that's where laws of chemistry and biology come in.
You say that "Also, in all cases, the probability that the average of all the numbers will be exactly at the center of the bell curve approaches 100% as the number of samples gets bigger and bigger.". Ain't this wrong? To me, it seems like the probability of being exactly in the centre is always equal to zero when choosing a real number between two values and tends to zero in the discrete case with coin flips.
