How chance affects our lives way more than you think | The mathematics of randomness 

kev luo I definitely remember my favorite triangle

DB Editing my favourite triangle is an equilateral

@@justaderrickrosestan mmmmmm

Nl

Merci fcfv

instablaster.

yea same random looks less random than it is

Good call

Right? I listen to his videos while driving, and the dude is so good at explaining that im able to grasp the concepts clearly.

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.

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?

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.

^

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

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)

Can confirm this is correct, I was the shark.

If anyone has a link to that method and the theory behind it, I would love to read it

Yah seriously

no

Amazing comment, it seems that you've learned the lesson well even though I've set up my S_threshold high 👍

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).

videoblocks.com

MajorPrep ok cool.

You can't

I know, very hard, but at least pseudorandom at its best

Thats a very expectable outcome

This is called the Central Limit Theorem, just in case you wanna sound really smart =)

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.

1 - (probability of getting 0, 1, 2 and 3 heads)

I have no idea what you're saying. How did you even come up with the 50%? Your subjective opinion on the likelihood of an event bears no relevance to its probability. If probability of X is 60%, then it will have a 60% chance of happening.