What concepts or facts do you know from math that is mind blowing, awesome, or simply fascinating?
Here are some I would like to share:
- Gödel's incompleteness theorems: There are some problems in math so difficult that it can never be solved no matter how much time you put into it.
- Halting problem: It is impossible to write a program that can figure out whether or not any input program loops forever or finishes running. (Undecidablity)
The Busy Beaver function
Now this is the mind blowing one. What is the largest non-infinite number you know? Graham's Number? TREE(3)? TREE(TREE(3))? This one will beat it easily.
- The Busy Beaver function produces the fastest growing number that is theoretically possible. These numbers are so large we don't even know if you can compute the function to get the value even with an infinitely powerful PC.
- In fact, just the mere act of being able to compute the value would mean solving the hardest problems in mathematics.
- Σ(1) = 1
- Σ(4) = 13
- Σ(6) > 10^10^10^10^10^10^10^10^10^10^10^10^10^10^10 (10s are stacked on each other)
- Σ(17) > Graham's Number
- Σ(27) If you can compute this function the Goldbach conjecture is false.
- Σ(744) If you can compute this function the Riemann hypothesis is false.
Sources:
- YouTube - The Busy Beaver function by Mutual Information
- YouTube - Gödel's incompleteness Theorem by Veritasium
- YouTube - Halting Problem by Computerphile
- YouTube - Graham's Number by Numberphile
- YouTube - TREE(3) by Numberphile
- Wikipedia - Gödel's incompleteness theorems
- Wikipedia - Halting Problem
- Wikipedia - Busy Beaver
- Wikipedia - Riemann hypothesis
- Wikipedia - Goldbach's conjecture
- Wikipedia - Millennium Prize Problems - $1,000,000 Reward for a solution
For the uninitiated, the monty Hall problem is a good one.
Start with 3 closed doors, and an announcer who knows what's behind each. The announcer says that behind 2 of the doors is a goat, and behind the third door is ~~a car~~ student debt relief, but doesn't tell you which door leads to which. They then let you pick a door, and you will get what's behind the door. Before you open it, they open a different door than your choice and reveal a goat. Then the announcer says you are allowed to change your choice.
So should you switch?
The answer turns out to be yes. 2/3rds of the time you are better off switching. But even famous mathematicians didn't believe it at first.
I know the problem is easier to visualize if you increase the number of doors. Let's say you start with 1000 doors, you choose one and the announcer opens 998 other doors with goats. In this way is evident you should switch because unless you were incredibly lucky to pick up the initial door with the prize between 1000, the other door will have it.
I now recall there was a numberphile with exactly that visualisation! It's a clever visual
It really is, it's how my probability class finally got me to understand why this solution is true.
This is so mind blowing to me, because I get what you're saying logically, but my gut still tells me it's a 50/50 chance.
But I think the reason it is true is because the other person didn't choose the other 998 doors randomly. So if you chose any of the other 998 doors, it would still be between the door you chose and the winner, other than the 1/1000 chance that you chose right at the beginning.
I don't find this more intuitive. It's still one or the other door.
The point is, the odds don't get recomputed after the other doors are opened. In effect you were offered two choices at the start: choose one door, or choose all of the other 999 doors.
This is the way to think about it.
The thing is, you pick the door totally randomly and since there are more goats, the chance to pick a goat is higher. That means there's a 2/3 chance that the door you initially picked is a goat. The announcer picks the other goat with a 100% chance, which means the last remaining door most likely has the prize behind it
Edit: seems like this was already answered by someone else, but I didn't see their comment due to federation delay. Sorry
Don't be sorry, your comment was the first time I actually understood how it works. Like I understand the numbers, but I still didn't get the problem, even when increasing the amount of doors. It was your explanation that made it actually click.
I think the problem is worded specifically to hide the fact that you're creating two set of doors by picking a door, and that shrinking a set actually make each individual door in that set more likely to have the prize.
Think of it this way : You have 4 doors, 2 blue doors and 2 red doors. I tell you that there is 50% chance of the prize to be in either a blue or a red door. Now I get to remove a red door that is confirmed to not have the prize. If you had to chose, would you pick a blue door or a red door? Seems obvious now that the remaining red door is somehow a safer pick. This is kind of what is happening in the initial problem, but since the second ensemble is bigger to begin with (the two doors you did not pick), it sort of trick you into ignoring the fact that the ensemble shrank and that it made the remaining door more "valuable", since the two ensembles are now of equal size, but only one ensemble shrank, and it was always at 2/3 odds of containing the prize.
The odds you picked the correct door at the start is 1/1000, that means there's a 999/1000 chance it's in one of the other 999 doors. If the man opens 998 doors and leaves one left then that door has 999/1000 chance of having the prize.
Same here, even after reading other explanations I don't see how the odds are anything other than 50/50.
read up on the law of total probability. prob(car is behind door #1) = 1/3. monty opens door #3, shows you a goat. prob(car behind door #1) = 1/3, unchanged from before. prob(car is behind door #2) + prob(car behind door #1) = 1. therefore, prob(car is behind door #2) = 2/3.
Following that cascade, didn't you just change the probability of door 2? It was 1/3 like the other two. Then you opened door three. Why would door two be 2/3 now? Door 2 changes for no disclosed reason, but door 1 doesn't? Why does door 1 have a fixed probability when door 2 doesn't?
No, you didn’t change the prob of #2. Prob(car behind 2) + prob(car behind 3) = 2/3. Monty shows you that prob(car behind 3) = 0.
This can also be understood through conditional probabilities, if that’s easier for you.
How do we even come up with such amazing problems right ? It's fascinating.
This is fantastic, thank you.
They emphatically did not believe it at first. Marilyn vos Savant was flooded with about 10,000 letters after publishing the famous 1990 article, and had to write two followup articles to clarify the logic involved.
Oh that's cool - I had heard one or two examples only. Is there some popular writeup of the story from Savant's view?
I couldn't tell you - I used the Wikipedia article to reference the specifics and I'm not sure where I first heard about the story. I just remember that the mathematics community dogpiled on her hard for some time and has since completely turned around to accept her answer as correct.
Also relevant - she did not invent the problem, but her article is considered by some to have been what popularized it.
I know it to be true, I've heard it dozens of times, but my dumb brain still refuses to accept the solution everytime. It's kind of crazy really
Let's name the goats Alice and Bob. You pick at random between Alice, Bob, and the Car, each with 1/3 chance. Let's examine each case.
Case 1: You picked Alice. Monty eliminates Bob. Switching wins. (1/3)
Case 2: You picked Bob. Monty eliminates Alice. Switching wins. (1/3)
Case 3: You picked the Car. Monty eliminates either Alice or Bob. You don't know which, but it doesn't matter-- switching loses. (1/3)
It comes down to the fact that Monty always eliminates a goat, which is why there is only one possibility in each of these (equally probable) cases.
From another point of view: Monty revealing a goat does not provide us any new information, because we know in advance that he must always do so. Hence our original odds of picking correctly (p=1/3) cannot change.
In the variant "Monty Fall" problem, where Monty opens a random door, we perform the same analysis:
As you can see, there is now a chance that Monty reveals the car resulting in an instant game over-- a 1/3 chance, to be exact. If Monty just so happens to reveal a goat, we instantly know that cases 1b and 2b are impossible. (In this variant, Monty revealing a goat reveals new information!) Of the remaining (still equally probable!) cases, switching wins half the time.
To me, it makes sense because there was initially 2 chances out of 3 for the prize to be in the doors you did not pick. Revealing a door, exclusively on doors you did not pick, does not reset the odds of the whole problem, it is still more likely that the prize is in one of the door you did not pick, and a door was removed from that pool.
Imo, the key element here is that your own door cannot be revealed early, or else changing your choice would not matter, so it is never "tested", and this ultimately make the other door more "vouched" for, statistically, and since you know that the door was more likely to be in the other set to begin with, well, might as well switch!
like on paper the odds on your original door was 1/3 and the option door is 1/2, but in reality with the original information both doors were 1/3 and now with the new information both doors are 1/2.
Your original odds were 1/3, and this never changes since you don't get any new information.
The key is that Monty always reveals a goat. No matter what you choose, even before you make your choice, you know Monty will reveal a goat. Therefore, when he does so, you learn nothing you didn't already know.
Yes, you don't actually have to switch. You could also throw a coin to decide to stay at the current door or to switch. By throwing a coin, you actually improved your chances of winning the price.
This is incorrect. The way the Monty Hall problem is formulated means staying at the current door has 1/3 chance of winning, and switching gives you 2/3 chance. Flipping a coin doesn't change anything. I'm not going to give a long explanation on why this is true since there are plenty other explanations in other comments already.
This is a common misconception that switching is better because it improves your chances from 1/3 to 1/2, whereas it actually increases to 2/3.
This explanation really helped me make sense of it: Monty Hall Problem (best explanation) - Numberphile
It's a good one.
It took me a while to wrap my head around this, but here’s how I finally got it:
There are three doors and one prize, so the odds of the prize being behind any particular door are 1/3. So let’s say you choose door #1. There’s a 1/3 chance that the prize is behind door #1 and, therefore, a 2/3 chance that the prize is behind either door #2 OR door #3.
Now here’s the catch. Monty opens door #2 and reveals that it does not contain the prize. The odds are the same as before – a 1/3 chance that the prize is behind door #1, and a 2/3 chance that the prize is behind either door #2 or door #3 – but now you know definitively that the prize isn’t behind door #2, so you can rule it out. Therefore, there’s a 1/3 chance that the prize is behind door #1, and a 2/3 chance that the prize is behind door #3. So you’ll be twice as likely to win the prize if you switch your choice from door #1 to door #3.
First, fuck you! I couldn't sleep. The possibility to win the car when you change is the possibility of your first choice to be goat, which is 2/3, because you only win when your first choice is goat when you always change.
x1: you win
x2: you change
x3: you pick goat at first choice
P(x1|x2,x3)=1 P(x1)=1/2 P(x3)=2/3 P(x2)=1/2
P(x1|x2) =?
Chain theory of probability:
P(x1,x2,x3)=P(x3|x1,x2)P(x1|x2)P(x2)=P(x1|x2,x3)P(x2|x3)P(x3)
From Bayes theorem: P(x3|x1,x2)= P(x1|x2,x3)P(x2)/P(x1) =1
x2 and x3 are independent P(x2|x3)=P(x2)
P(x1| x2)=P(x3)=2/3 P(x2|x1)=P(x1|x2)P(x2)/P(X1)=P(x1|x2)
P(x1=1|x2=0) = 1- P(x1=1|x2=1) = 1\3 is the probability to win if u do not change.
Why do you have a P(x1) = 1/2 at the start? I'm not sure what x1 means if we don't specify a strategy.
Just count the number of possibilities. If you change there there two possible first choices to win + if you do not change 1 possible choice to win = 3. If you change there is one possible first choice to lose + if you do not change there two possible first choices to lose=3 P(x1)=P(x1') = 3/6
Ah, so it's the probability you win by playing randomly. Gotcha. That makes sense, it becomes a choice between 2 doors
Without condition would be more technically correct term but yes