Big math/science results within easy reach

[JRS]

Sorry, looked up the info while you guys were posting. Edited my post to add the info before seeing the other posts. So we sort of posted the same stuff. That is okay however.

No worries. To reach 99% only requires 70.
Meanwhile the odds of 318 before having a match would edited big blunder on my part
Should be something like
(363!/45!)/364^318 or really effing small.

 

[pseudoid]

I thought you'd point out leap year.

If I remember correctly, the omission was intentional!

 

[Hatto]

I thought you'd point out leap year. Someone born on February 29th, and you could have 366 and no one with the same birthday.

You walking into a room with 366 others make it 367, that make it certain including leap year condition.

 

[RayDunzl]

A bit of Zeno's paradox about the time taken to for the arrow to go 1/2 the distance to its target, and in the next interval to go 1/2 of that and so forth such that it takes infinite time to reach the target.

Not if the arrow maintains the same speed.

The fractional distances left to go blow up real good, though.

If you disagree, show me the arrow that (well aimed and with appropriate velocity) never hits the target.

I suppose the paradox blows up when the arrow gets below a Planck length from the target, because it can't, so it's there.

 

[Blumlein 88]

No worries. To reach 99% only requires 70.
Meanwhile the odds of 318 before having a match would edited big blunder on my part
Should be something like
(363!/45!)/364^318 or really effing small.

We also had a math teacher who was really sharp. Don't know why she was a teacher and after two years IBM hired her for mathematical research. The physics guy talked to her to find out if he was missing something. She had a class of 28 juniors at the time and two had the same birthday. Her next class of about 25 had three with the same birthday. So a little above normal, but more in line with expectations. So yes 318 is nearly impossible, but that is improbable not impossible.

So yes, really effing small. It is quite possible, though not guaranteed that there is no other high school class of around that size ever that didn't have any duplicate birthdays. Nor may there ever be again.

 

[jsilvela]

I'd say that Godel's theorem is pretty approachable. Pretty much the Cantor diagonal proof on steroids.

Amazon.com

View attachment 278398

(above is a pointer to Amazon : "on formally undecidable propositions of the principia mathematica"

A bit more work is required to digest the Pea and the Sun paradox (Banach-Tarski Paradox), but once there your mind will be properly blown as the realization of how strange infinite sets can be hits home.

Amazon.com

have fun!

jack

I've never gone full into Godel, but was aware that at the core, it was Cantor diagonal in disguise.
I may take the leap on the book then.

On Banach-Tarski. There's an easily reachable result that is in a similar vein: Vitali's construction of a non-measurable set.

Step 1: define what you mean by "measure" ... really simple a function m such that m(A union B) = m(A) + m(B) for disjoint sets, and two simple rules more.
Step 2: create a set that cannot be measured with the function above, no matter how it's implemented.

Like Banach-Tarski, the Vitali set requires the Axiom of Choice.
Makes you wonder.

 

[jsilvela]

Not if the arrow maintains the same speed.

The fractional distances left to go blow up real good, though.

If you disagree, show me the arrow that (well aimed and with appropriate velocity) never hits the target.

I suppose the paradox blows up when the arrow gets below a Planck length from the target, because it can't, so it's there.

There is something to Zeno's paradox though.
The conclusion that it takes infinite time to hit the target is not right.
But realizing you can divide the journey into an infinite series of intervals, each 1/2 the previous one's length. That there is an infinity contained in the journey is kind of mind-boggling, even if today a high-school student will easily walk past this hurdle.

Not Zeno's paradox at all, but I still find it difficult to wrap my head around this:
There is no [real/rational] number that is the largest one less than 1.

 

[jsilvela]

A bit of Zeno's paradox about the time taken to for the arrow to go 1/2 the distance to its target, and in the next interval to go 1/2 of that and so forth such that it takes infinite time to reach the target.

Actually, if we use binary notation, where the "decimals" denote 1/2, 1/4, 1/8 ... rather than 1/10, 1/100, 1/1000 as in base 10,
then Zeno's paradox is "solved" by saying 0.1bar = 1

 

[RandomEar]

Statistics offer a number of counter-intuitive but totally logical revelations. Two of my favourites:

• Simpson's paradox
  Essentially, a statistical correlation for one variable derived from a complete data set can be totally different from the correlation of the same variable derived from sub-sets of the original dataset. The visual example given on Wikipedia makes this very easy to grasp. This is a very common problem with many "human-related" statistics (medicine, social sciences, and so on).
  It's a dangerous fallacy, especially because people not versed in statistics like to confound correlation with causation. If you get the wrong result from an analysis, then mix up cause and effect and then act on that misunderstanding and possibly create laws, policies or make important decisions, you are likely to act against your own interests and make things worse.
• Monty Hall Problem
  The famous "game show paradox": You are in a game show and need to choose one of three doors. A a price is hidden behind one of them. You pick a door and the show host then deletes one of the two other doors from the game, stating that that one was a dud. You are then given the chance to update your original choice. Your chances of winning the price increase, if you take that chance and switch doors.
  This - at least for me - was counter-intuitive at first glance. But the numbers don't lie: You do improve your chances by switching, mainly because you have gained some information when the host deleted one of the doors following your first selection.

 

[IAtaman]

Not if the arrow maintains the same speed.

The fractional distances left to go blow up real good, though.

If you disagree, show me the arrow that (well aimed and with appropriate velocity) never hits the target.

I suppose the paradox blows up when the arrow gets below a Planck length from the target, because it can't, so it's there.

There are 2 weird things about that fact in my opinion. One, do we live in a discrete universe, and two, even if we live in an continuous universe that sum adds up to one, so how come math is so incredibly and unreasonably effective at explaining real world. Here is a nice paper on the latter : https://www.maths.ed.ac.uk/~v1ranick/papers/wigner.pdf

 

[IAtaman]

• Monty Hall Problem
  The famous "game show paradox": You are in a game show and need to choose one of three doors. A a price is hidden behind one of them. You pick a door and the show host then deletes one of the two other doors from the game, stating that that one was a dud. You are then given the chance to update your original choice. Your chances of winning the price increase, if you take that chance and switch doors.
  This - at least for me - was counter-intuitive at first glance. But the numbers don't lie: You do improve your chances by switching, mainly because you have gained some information when the host deleted one of the doors following your first selection.

Oh that is some dangerous stuff you got over there. That can end friendships! In fact it can even be a anti example to the thread, in that, it is very easy to contruct the problem and very hard to grasp why it works the way it does for a human brain that has evolved to survive in equatorial grasslands.

 

[BDWoody]

Statistics offer a number of counter-intuitive but totally logical revelations.

I don't know if it is exactly counter-intuitive, but using the example of a deck of playing cards and the factorial function can be fun. The uniqueness of a random shuffle, for example, and the likelihood that any two decks of cards have ever been shuffled the same way, or ever will be. 68 digit numbers can be hard to get your head around.

 

[mhardy6647]

68 digit numbers can be hard to get your head around.

Fortunately.
Our ancestors probably woulda climbed right on back up into the trees if they'd have thought too hard about 68 digit numbers*.

_____________
* especially when the one in their clan with a pocket protector and crude, horn-rimmed glasses with very thick lenses grunted: "oog! What about one with sixty-nine digits, or seventy?"

I know how they'd have reacted.

 

[BDWoody]

* especially when the one in their clan with a pocket protector and crude, horn-rimmed glasses with very thick lenses grunted: "oog! What about one with sixty-nine digits, or seventy?"

"What?! We've got wild cards and jokers in this deck?!"

 

[mhardy6647]

"What?! We've got wild cards and jokers in this deck?!"

Thank goodness they didn't have roulette wheels yet!

 

[tomtoo]

I hate statistics.

I have a coin with a one or two on it. 50% that the 1 comes 50% that the 2 comes.

Now i play around with it, and 15 times the two came out. The probabylity is still 50% for both. And thats the strange part.
Would i bet on the one or the two now?

 

[RayDunzl]

I hate statistics

To me. statistics tells you how what just happened falls into the previously calculated range for stuff that happens, but tells you very little if anything about what is about to happen.

Heads or tails next?

Nobody knows and the statistics of what happened before won't tell you.

 

[tomtoo]

To me. statistics tells you how what just happened falls into the previously calculated range for stuff that happens, but tells you very little if anything about what is about to happen.

Heads or tails next?

Nobody knows and the statistics of what happened before won't tell you.

Its hard to get for me. Somehow it tells there is now past, and what happens now is completly independent from what happend in the past. But on the other hand if you see 20 times head, you wont bet on it. Couse the more often you have seen head the higher should the probabylity for tail to fit the 50:50. Its somehow a paradox for me. Even i know its 50:50 always.

 

[RandomEar]

Oh that is some dangerous stuff you got over there. That can end friendships! In fact it can even be a anti example to the thread, in that, it is very easy to contruct the problem and very hard to grasp why it works the way it does for a human brain that has evolved to survive in equatorial grasslands.

If you've always wanted to end a specific friendship, I can additionally recommend the sleeping beauty problem

 

[tomtoo]

I did watch it some days ago. Than i thought should i think about it, or drink a beer?
I told you i hate statistics.