How chance affects our lives way more than you think | The mathematics of randomness
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- Опубліковано 21 вер 2024
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This was way more fascinating than I had imagined, especially the changepoint calculations. Great vid!
instablaster.
3:29 "I'm sure many of you know that this is actually the fifth row of pascals triangle"
Ah yes of course I knew that... Haha..........
kev luo I definitely remember my favorite triangle
DB Editing my favourite triangle is an equilateral
@@justaderrickrosestan mmmmmm
Nl
Merci fcfv
What is the probability that the next video is also probably going to be explaining the intricacies of probability?
"Oh, never mind!"
I guessed correctly, reasoning that that one ended with 6 Heads in a row, which a human trying to mimic randomness would never do.
yea same random looks less random than it is
Good call
I don't usually comment on any video but I feel obligated to thank you for your amazing content. Although I am not much of a STEM guy, I absolutely love probability/statistics and the way you present these themes on your videos is simply magical!! Greetings from Brazil
Right? I listen to his videos while driving, and the dude is so good at explaining that im able to grasp the concepts clearly.
That channel is really interesting as a student in the university (first year)
Keep posting this probability videos!!
Im also a freshman :D
Hey I like to play the lotto. I pick my numbers but I don't use them cause more than likely I'm wrong so I throw those out. Do u have any ideas?
@@jakegarza4131 if we knew then we'd be winning to lotto.
Statistics and stochastic are often overlooked while they are so unbelievably important, especially today. I wish I had acquainted myself with both much earlier.
I just binge watched a bunch of your videos and I feel so smart lmao
This was a super cool video, I've never thought of probability and randomness to affect us in this way, and I was actually sad that the video wasn't longer. Well done :)
So, randomness aka luck is an universal phinomina.
So philosophically speaking, we should always be hopeful for the best ☺ .. and when we get something we dream of, we shouldn't be arrogant of out handwork (role of hardwork aka trial always present, I don't deny that) , knowing the fact "we are lucky"
This video has inspired me to spend all my time at the casino, because if I try long enough eventually I'll hit it big
I really love your content, its something I watch while having lunch/dinner. Please keep making them.
Dude you're fucking brilliant. Your understanding and explaining of these concepts is so strong.
Great video! randomness is a very interesting topic, well when you relate it to things like chaos theory and fractals
A decade ago, there was incontrovertible proof that Newyorkers don't understand randomness (or most Newyorkers don't). A cancer hot spot was detected in a Long island town and all hell broke loose because it was above average. The townsfolk proceeded to fund studies that blamed, the water, the nearby Brookhaven Nuclear Lab, the new high tension electric wire and a whole bunch of other stuff. They just refused to believe that they were unlucky while the other towns were at average or below. They really did think that random meant even distribution with slight variations.
11:25
"Bio-terrorist attack"
Man, this did not age well...
Yesss that first example I first saw in sixth grade when my geometry teacher talked about a study with this exact result. And then saw again while I was moderating a topic on aops class message board lol
Technically , South Africa's shark attacks in 2010 could be because of the world cup being held there . Meaning beaches being one of its tourism sites , more tourists
would want to swim ,thus increasing the number of people who swim per day and that would increase the chance of more shark attacks
But what can I say am just a kid
Can confirm this is correct, I was the shark.
Having flipped a coin 334 times in a row a few days before watching this, that was a super easy guess
If only someone was as well informed about change point when it came to warning about COVID
First of all great content, really fascinating! I have a request from you that you please also share name of papers that you said you refer to in your video while researching. That way more interested viewers of your can also take a read and have a general great conversation about the topic.
0:57 Its fun to pause these videos before you reveal the answer and try to figure out it came to be!
Thanks
Now I only need to optimize my risk aversion for success; taking too few risks can leave a person in a rut of unsuccessfulness, but deciding not to avert any risks at all could prove to be unhealthy.
Add to your expectations the positive knowing that outcome is often influenced totally by random variables. To me, this makes life interesting. Then failures are not seen as 'all my fault.' Reading bios of famous entrpreneurs or artists, I've learned that many of them ended their lives in poverty and obscurity. Their wealth and recognition were not directly related to their efforts...though recognition came later.
But we have to be content within our own mind, knowing we can't control all the variables or other people...or evil. With this attitude, life can be seen as one big adventure. (and another huge piece to the puzzle is faith and the interruption of random by divine intervention)
That was great MP, easily the best one I’ve seen. You’re really starting to hit your stride with this 😎👍
I love your videos. Keep up the great work.
Hey, I just found this channel and it's amazing! Keep up the good work, subscribed.
Fun content! Randomness is really important because it includes more than what already is.
One of the best youtube videos I have seen. Thanks!
Woah how did Pascal’s triangle get involved when modeling coin flips?
Simple. A number in Pascal's triangle is the number of paths to that point from the top of the triangle. At each point in the triangle, there is a binary choice for where to go next: down to the left and down to the right. Start at the top of the triangle and choose the direction to go by flipping a coin. Since the path to any particular number in the triangle always involves the same number of left and right steps, just in different orders, that number corresponds to the number of ways of getting a certain number of heads and tails in any order. That gives you the numerator of the probability, and the denominator is the sum of the numbers in that row (which is the total number of possible outcomes for a certain number of coin flips).
Paul Giaccone Beautiful 🙌🏽
You also get (a+b)^n using Pascal’s triangle assuming you label the first row as row 0
Paul Giaccone “Simple” *proceeds to have a long explanation that I don’t understand*
@@mrcarlwheezer584 Hopefully simpler explanation :
I'm trying to find the number of ways that I can flip coins and get 2 heads and 2 tails. I can view this as starting with 4 heads, and choosing two of them that I'll change to tails. The number of ways to get 2-2 is the number of ways to choose 2 elements from a set of 4 elements, or "4 choose 2".
Refresher course on Pascal's triangle : Pascal's triangle can be thought of as a right triangle 📐 where the value of a cell is the value of the cell directly above + the value of the cell above and to the left. Calling the cell (n,k), n being the row and k being the column, we get (n,k) = (n-1,k) + (n-1,k-1).
Now how are the two related ? The answer is surprisingly simple. It turns out that "n choose k", or the number of ways that you can choose k elements from a set of size n, is the value of the cell (n,k). To see this, imagine a list of n elements : (1), (2), (3)... (n).
You have to pick k of them. Think about the first one : either you pick it or you don't. If you pick it, you'll have to pick k-1 elements from the n-1 others. So the number of ways to choose k from n while picking the first element is equal to the number of ways to choose k-1 from n-1. If you don't pick it, you'll still have to pick k elements, but you're limited to n-1. Thus "n choose k" = "n-1 choose k" + "n-1 choose k-1". Considering you have only one way to pick none or all of them (don't have much choice there in which ones you pick), "n choose 0" = "n choose n" = 1. You have the same recurrence relation and the same base values, so (n,k)
= "n choose k".
So the number of ways to get a certain number of heads and tails out of n throws is the nth row of Pascal's triangle.
An absolutely amazing book, which talks about randomness, specifically about the stock market is "Fooled By Randomness", I suggest giving it a read.
I would be interested in more information about how the false positive rates and detection times are determined from the threshold selection.
If anyone has a link to that method and the theory behind it, I would love to read it
In your crime-increase detection system, could taking the derivative of S also provide some information. For example, if it crosses your threshold, monitoring the derivative for a few days may help discriminate false positives quickly?
If there was an option there to like this video twice, I would!
Context: I give videos a Thumbs Up (👍) less but this and Slayy-Point is the only Channel I've thumbed whom videos so much of times
I want to thank you for existing!!!!
Amazing how 17 minutes can change your perception of randomness.
12:16 you said to assume the daily probability of a crime happening if there is one crime a month on average would be 1/30 for every day, but what if a crime is more likely to happen on a Sunday, or a Friday, or what if there was a blizzard outside and there would be nobody to commit any crimes?
Apparently apple had a shuffle feature on their iPod and they had to make it less random due to complaints that it “wasn’t random enough”. People were getting the same songs multiple times in a row. Their algorithm for shuffling songs was too efficient and random it actually made it appear less random to people who wanted a new song every skip
My friend thinks he has better hearing than me, but really his auditory cortex just has a lower S-threshold.
Amazing comment, it seems that you've learned the lesson well even though I've set up my S_threshold high 👍
Just little thought experiment:
If I go to a casino and bet 100$ on the roulette black/red, if I win I walk home having earned money, if I lose I bet 200$, so that ill have earned 100$ total after winning, since I lost the initial 100$.
If I lose the second I play a third time for 400$, and so on, shouldnt I expect to come home richer, since the probability is 50/50 (not exactly but close) on each throw so this chance of having lost money after 10 attempts for example is 1/1024, assuming 50/50 of course, which again I know is incorrect, but for the sake of the thought experiment
Every test I run on my state lottery results is right in line with expected probabilities.
Dude can you please make a video on nuclear physics/ Nuclear Engineering..
Yah seriously
Underrated channel.
You example about comparing crime rates from year to year is really important. A lot of people like to cite a study done in Canada that found that after a small anti-rape poster campaign the number of rapes in the city dropped by 10%. That sound impressive, until you realize that from year to year the number of rapes randomly went up or down by about 10% regardless of whether there was an anti-rape poster campaign. People expect crime to be relatively static and then attribute any change to whatever policy was enacted most recently, when in actuality it constantly goes up and down for basically no reason. Even if you have a couple of back to back exceptional years that doesn't really prove much because in the grand scheme of things it's not that remarkable.
How doesn't this channel have more likes
Poisson has a probability mass function, not density.
11:40 Do you have a link/paper/anything to that simplified algorithm you talk about? Or is it in the book you mention in the description?
That one came from the "Numbers behind numb3rs" book. There's pretty much an entire chapter dedicated to it. The simpler algorithm is from someone named E.S. page. The book also mentions a more complex algorithm called the Shiryayev-Roberts method which was the one mentioned in that episode of Numb3rs. The math in any source I found on that is pretty intense.
@@zachstar Thanks! Really great videos/channel by the way.
Google statistical process control, has some similarities to this.
I believe this Wikipedia article covers it: en.wikipedia.org/wiki/CUSUM
Zach Star For practical purposes, is it possible to say that the likelihood of a chance outcome would be zero? Example: suppose you have 10,000 dice in a box, and you dump them out at once. What are the odds of them all coming up 6?
Without doing the actual math, suppose the likelihood of that outcome would take a billion people rolling their dice over and over again for a billion years. Would that be a probably of zero for a real world situation?
The details have all gone fuzzy but back in the 80's I remember using the chi squared test to determine if a sequence of values was likely random or not. I was in an agronomy major at the time and there was tons of work put into analyzing results from experiments. I still have bad memories of the SAS application and their ANOVA (analysis of variance) module. ;)
What's The Chance, We develop beyond)( our Six Senses .Yes.
Honestly, can you really provide an example of anything that is truly random? The disconnect between cause and effect and our ignorance and inability to account for all factors creates a scenario where random is simply a lack of information. Another concept we made up to explain things we can't know. The difference between someone who is able to predict outcomes and everyone else is they simply pay attention to relevant data. The more information you're able to gather and process, the less meaning "random" has.
I like how you looked at the 4 heads in a row and not the multiple sets of 6
...just wow!!
nothing is completly random, its always backed up by either the laws of physics or actions of other human beings, or a combination of both. no matter what happens, luck can always be backed up by logical reasoning.
Please make a video on William Lowell Putnam Mathematical Competition
so what youre saying is that 60% of the time it works every time?
I was trying to do something last week which involved trying to figure out the area of something that grows exponentially and I had a hard time figuring it out. But I eventually stopped, started over and came up with something that looks like pascals triangle.
I've heard of pascals triangle but I never bothered looking it up, so I didn't know what it was.
the eleventh row of pascals triangle is 2^10 = 1024
I realized this because each row starts and ends with the number 1
the value of the first row is 2^0 =1
the second row 2^1 = 2
the 3rd is 2^2 = 4
then 2^3 = 8
The total value of all the rows beneath the current row is the value of the current row minus 1
I can't remember exactly why I started doing this, but I started by growing a binary tree from the first element which splits into 2 children and each of those children split into 2 and so on. But I did it on paper and I tried to compactify it using numbers so that I can try to figure out a mathematical way to do math without doing all the math (so essentially a shortcut for solving exponential equations).
My idea didn't work but apparently I found pascals triangle on accident.
That tradeoff is the real concern. Math is perfect, but humans decide the treshold.
basically learning some AP statistic concepts in an entertaining way
Do a video on petroleum engineering.
Amazing content!
i love this channel
If you flip a coin,there is 50% chance the coin
to be Heads and Tails.This applies every time
you flip the coin.You could flip the coin a
hundred times altogether.One can say there
should be 50 Heads and 50 Tails.This might
not be the case.We have no control of
the outcome.In mathematics especially
Calculus we can apply parameters to
get the Solutions.
I am sure you have heard of the Equation
that the Angle of Dangle = the Square Root
of the Hole.I could never solve this Equation.
❤️.
I would argue that you can not compare the amount of crime in a city of 100 people with the amount of shark attacks in South Africa as the population in South Afrika is 560,000 times bigger. The chance of a citizen of being in a shark attack in 2010 has only increased by les than 0.00000001% compared to the average year. At the same time the chances of being involved in crime in the little city has increased by 4,5%.
I am therefore still convinced that there is something fishy going on in the town.
Calculations:
8*(1/100)-3.5/100 = 0.04500000000
8*(1/56000000)-3.5/56000000 = 8.035714290 * 10 ^-8
Great video!
If it's 50/50 to get H or T, what 'force' causes the adherence to that percentage, say, out of 1,000 flips? If I get TTTTTT, who says (what 'force') eventually causes enough Hs by 1,000 to equal it out?
A term that is demonstrated by events in life: anti-random. (which is not the same as patterns)
anti-random
would you explain in a new video the statistic behind a clinical trials? thanks!!
Intresting, I didn't know about this!
I got the first question wrong, so I should have picked the lumpy sample, but it just seemed too lumpy.
Nice video! Could you link to the paper where you found the changepoint algorithm you just described?
^
I got that one from a book actually which is linked in the description. I found this link for it though which contains everything I said. books.google.com/books?id=YN2el7wmTFIC&pg=PA56&lpg=PA56&dq=e.s.+page+changepoint+detection+numbers+behind+numb3rs&source=bl&ots=eReInhz6y9&sig=ACfU3U1QPlaZJFSbReAb82QKyB2_ZmCUuQ&hl=en&sa=X&ved=2ahUKEwj8hODEmt7hAhXCwFQKHZtBBykQ6AEwAHoECAgQAQ#v=onepage&q=e.s.%20page%20changepoint%20detection%20numbers%20behind%20numb3rs&f=false
This is some beautiful math! How do you calculate the false alarm numbers at timestamp 14:20?
Funny thing is, if I may add, is nothing is truly random, no matter what, if you look microscopically enough.
For example, at 6:35 you say "is this increase due to some external factor, or randomness?" In reality, what spurs people to perform crimes has a reason behind it. And these reasons are basically what you mean by "some external factor" as long as you look microscopically enough, but it's just not humanely practical to factor in these factors.
Another example is throwing a dice. If you could hypothetically throw a dice the exact way and the surface it lands on is constant and wind is non-existent, etc etc, it's not going to be random what number comes up on the dice.
It's just that a tiny input creates a result different enough that this is perceived as randomness -- because those tiny inputs can never be humanely factored in, as they happen on the atomic level or even smaller in the near-infinite magnitude of instances per unit time.
The fluidity in the perception of randomness once you understand this is what is truly mind-blowing -- exactly what draws the line on thus microscopic level of "factors"? Where is it untouchable and where is it controllable, truly?
On the side I built a gambling system consists of higher maths
I'm thinking RANDOM things right now...
*Wait*
Thats a very expectable outcome
But what if it has a 50% probability? So (1/2)÷(1/2) and (1/2)÷(1/2)? It either case we multiply 1*1 and it stays at 1.
Or do we actually after this rearange the probability? If we have 100 datapoints that give us 50%, after the new one we probably have 51% or 49%. And then add the outcome we had befor?
edit: say for example if we have 200 stocktrades as data pointa and want to see if a certain strategy doesn't work anymore.
edit2: ok i had an error. 1x a month / 1x a week isn't 50÷50, it's actually just the "new peobability".
Thanks phil
8:33 what if it is bigger probability if more crimes are caused by external factor that it is by a random chance
Everytime someone says it can't be random, it has a pattern, it most likely is random.
Why does this guy make me want to learn math even more? 😅 I want to know more @majorprep... how can I learn all this math and more on my own? I also want to learn more physics, electrical, computer and mechanical engineering... Any advice? Any online resources I can use (I am still young, I haven't gone to Uni yet, and I don't believe in uni teach me what I want to learn, so please help me out)?
What happens in the subatomic level when you are making decisions? How random mathematics can predict such complex mechanics?
"Unless you sample everyone in a nation or a city or whatever, which typically isn't possible"
me: *that's what a vote is*
How do you calculate the probability of a string of 4 heads coming up in a sequence of 100 flips?
A chain of 4 heads can start on days 1 - 97 (totalling 97 days).
On each one of these days, there is a 6.25% chance of there being a chain of four heads (100%*[1/2]^4=6.25%)
or a 93.75% chance of there not being a chain of four heads (100%-6.25%=93.75%).
There is one case of there being no chain of four heads, with each other case having chains of four heads.
It is much simpler to calculate the probability of there being no chains and subtract that from 100% than to calculate the probability for each case, so we'll do that.
Multiply the chance of no chains occurring by the number of days it can start on, and you get about 0.191% chance of no chains of four heads (93.75%^97=0.191%).
Subtract that from 100% to get to a final answer of 99.809% chance of a chain of four heads occurring (100%-0.191%=99.809%).
Note: It seems that my answer is off by 0.1, so the problem is probably more complicated than this; hopefully it still helps you get the general idea!
Catraption These tosses were not done 1 toss per day. I don’t know where you got the idea from. And is it 93.75%^97 or 93.75%*97? Also the 6.25% only accounts for 4 flips.
Why don't you use 0 before decimal point?? Drives me crazy.
i guessed it right.. not because i know the probability of 4 consecutive heads happening or something.. but the other one seems more like what i would have done if i wanted it to look random
What is the difference between randomness and chaotic?
Pure random is 50% possibility if there are only 2 possible outcomes.
So for pure randomness, you have to do any experiment only once.
As soon as you do experiment twice possibility changes from 50% to other value. For tossing coin, possibility is 50% for the first toss, but as soon as you toss again, possibility changes.
Any such random experiment can be done by many people in the world. Someone not knowing about the experiment has 50% possibility of an outcome, but otherwise possibility is not 50%.
As long as the random experiment is the same, we can compare results.
So pure random is just not knowing something. For example, we don't "know" whether toss will yield head or tail, but some supernatural entity can calculate force of toss and environmental conditions and can predict the outcome.
Sort of. Knowledge can affect the result of an experiment, but not all experiments are affected by whether the person performing them has knowledge. No matter how many times a regular human will toss a coin, he'll still get heads and tails roughly half the time, because while the first toss/first couple of tosses tells him some properties of the coin, this isn't enough for them to manipulate how the coin lands (because of gyroscopic stability, coins are almost impossible to "load" to a significant degree, unlike other objects like dice).
@@tudornaconecinii3609 People should do coin toss experiment by using a machine, in an empty room, and only should go in to check results of toss. Machine should be changed for each experiment too.
If successive tosses shows exact 50% probability, then we can safely say observation didn't affect experiment. Observing experiment is different than observing results.
Observation affecting experiment is also suggested by double slit experiment.
@@jaydeepvipradas8606 First of all, you'd get decoherence because of the sheer size of the entanglement at a macro scale. Your result would still be probabilistic, but it would lay within a much smaller margin of variation simply because of the number of entanglements involved (sort of like how the half-life of a chunk of pure Uranium is more consistent with the average than that of a tiny fleck of Uranium). And at the very least you'd need to empty the machine room of air because at that point you're entangling the coin-state with events miles and miles away.
Second of all, the level of technological and mathematical precision required if you wanted to make a machine that can predict the results of its own coin tosses would literally exceed that of launching and maintaining satellites in stable orbits around the Earth. Like, you could do it, but it's freaking harder than rocket science, so why waste time.
@@tudornaconecinii3609 Only constraint is that experiment should not be observed and no one should have knowledge of results.
Different person can observe the result with each coin toss without sharing it with anyone. Engagement with other elements in the universe can not be avoided even if we remove air, because vibrations will pay role. But entanglement is not constraint, knowledge and observation are constraints.
Machines should be different for each experiment and each machine should apply random amount of force on coin. Machines need not predict outcome of toss. Human person will go in the room to see coin result.
We are just trying to see whether successive coin tosses still gives 50% possibility of outcomes.
As per video, with observation and knowledge, after first toss, next toss doesn't have exact 50% possibility of outcomes, provided we know the first outcome and trying to find probability of next outcome.
I think, there is possibility of concluding some points.
When there are only 2 possible outcomes,
- For short number of experiments, say around 10, possibility is 50%, and we will see approximately same number of heads and tails.
- For large number of experiments, say 1000 and above, possibility is still 50%, and we will see approximately same number of heads and tails.
- For medium number of experiments, say 80 to 700, possibility is not around 50% and we will see heavy imbalance of number of heads and tails.
This is for 2 possible outcomes. When outcomes are more than 2, like dice, definition of small, large and medium experiments will change, I.e. what number is small, what is large and what is medium for number of experiments.
We may still be able to calculate these numbers by interpolation, by doing experiments of 3, 4, 5 etc outcomes.
This can be used for designing systems which involves randomness.
Who knows, system in casinos could have such bias and people think they have fair chance of winning.
When number of experiments are around medium level, patterns of outcomes will get developed, like HHHTH etc.
So to nullify the extra occurrences of an outcome, large number of experiments will have to be done to see 50% possible occurrence of each outcome.
Can you make a video about free will?
awesome
...how does the calculator random function work?
How come you have a magnetic levitation globe on your desk but not a Galton board? You know, the one they sell for $40 with 3,000 beads that always arrange themselves in a binomial but pretty darn close to normal distribution. They even come with Pascal's triangle and Fibonacci series printed on them. Vsauce did a whole video on the thing. I thought that it was a de riguer item for mathematicians.
Isn’t the chance of 4 consecutive flips being the same 12.5% since it doesn’t matter what the outcome of the first flip is?
the further you go down pascals triangle with that problem the more normally distributed the results
This is called the Central Limit Theorem, just in case you wanna sound really smart =)
keep the good work! this video is very interesting for me as a stats student since you explain what we can expect from a sample, I'd like to know wich software you used for plotting the poisson distribution also could you make map of statistics please? te amo no homo
Don't you hate it when you walk outside and someone throws a fire hydrant at you and then bounces on a trampoline while trying to eat you, and then another guy swoops in smacking your skull with his fist while shouting yEeHoO, and then some kid with a baseball bat shoots fire out of his fingers burning you half to death and then another kid walks up to you and literally freezes you in a small glacier, and then some ginger guy comes up with a giant sword and slams it down while shouting dOrIyAh, and then you hear the kid with the baseball bat from earlier continuously respond to your cries for help with oKaY and they all just walk away with your wallet AND your spine?
Happens every Saturday.
The right is the real flips I think
Good video
Can you do a Video about "Mechatronics Engineering" ?
I did a research... But just to be sure, it has to come from a guy with insight
6:48 Could you tell me what software or web site is that for calculating and displaying the poison distribution? Thanks.🙏🙏🙏
no
Hey guys. When he talked about the changepoint detection, in the example he gave some numbers on how often a false alarm will happen according to the threshold value. Maybe I'm just missing something, but how did he get those numbers on frequency of false alarms and time to detection?
Let me try to give a short, non-mathematical answer. I didn't double check the underlying paper, but that's how I'm sure it works:
In probability theory there's a thing called stopping time which measures if something has happened yet or not. You can use these stopping times and the algorithm to determine when/if the counter point occured (stopping time) and when the algorithm eventually indicates that it has. The average time in between (which is the expectation value of the difference between those stochastic processes) is what this model calls *detection time.*
*False alarms* happen when the threshold is reached without the process having stopped, you can determine the probability of this happening. With this you can use either the geometric or exponential distribution (discrete or continuous) to calculate how much time it takes on average to happen (for the first time).
Double slit experiment in practice.
Any reference on how to calculate the probability of getting minimum 4 heads out of 100 tosses ?
1 - (probability of getting 0, 1, 2 and 3 heads)
Random!
Hi! This question doesn't have something to do with this video but.. is it fine to take a degree in Electrical Engineer and then take a degree in Mechanical too? Let's say I'm planning to take Electrical engr now but I'm also planning to take mech engr after sometime I am already an electrical engineer