So you keep fliping a coin. 50/50 chances

[Matthias99]

Originally posted by: Skotty
Also, can you mathematically proove that .000...infinity...001 is equal to 0? I know it's easy to prove .9 repeating is equal to 1 by simply multiplying by 10 and cancelling out the .9 repeating. Through that you could claim that .9 repeating + .0...1 is equal to 1, and since .9 repeating is equal to 1, .0...1 must equal 0, but I'm not sure that is mathematically sound.

Mathematically, "0.0000...0001" is not a number (at least not a real or complex number). There's no way to actually define it. Logically, an "infinite number of zeroes followed by a one" cannot exist, since if there is a 1 at the end the zeroes would not actually be infinite.

"0.9999..." is actually "the limit as x goes to infinity of (the sum of i from 1 to x of (9 * (1/(10^i)))" -- that is, 0.9 + 0.09 + 0.009 + 0.0009 + ..., which is a well-defined infinite arithmetic series. This series is exactly equal to one.

The best you can do for "0.000...0001" is "the limit as x goes to infinity of (1/(10^x))". This limit is actually what you get when you take 1 and subtract (0.9 + 0.09 + 0.009 + ...) from it. That limit is exactly equal to 0 -- that's the way limits are defined.

 

[Agman]

ok I'm a probability and statistics class right now. And from what I can see if the coin is completely fair and you can replicate the enviorment perfectly each time in which is tossed. The probability that it will be heads in the first toss till be 50% like wise with heads. In the second toss since all the tosses are independent events (means that just because you got a head earlier it means you have a higher chance of getting a head now) the probability for heads and tails remains the same. If you think about it as a tree where each branch (heads or tails) branches off into 2 more branches( heads or tails) To get the probability of a certain combination you multiply the probability in each branch through the path you take to get your result. So in the end the probability of having infinite amount of heads will be lim as n-> infinity of (1/2)^n and since this is a geometric sequence in which the r value is less than 1 it converges. In this case it gets smaller and smaller converging to 0. So the probability of having an infinite amount of heads is pretty much 0 even though "realistically" there is an REALLY REALLY SMALL TINY MINUTE chance that it will happen.

 

[Skotty]

What's interesting is that each possible final result has the same really tiny minute chance that it will happen, not just the all heads or all tails scenario. So whatever result you get, you can be amazed that you got it, since it had such a small chance of happening.

 

[Agman]

yeah basically...if you take any combination to infinity it has the same exact minute chance mathematically of course because physically is impossible to have the same environment for an infinite amount of tosses...but interesting indeed

 

[jman19]

Originally posted by: DyslexicHobo
So is it zero? Or as close to zero as possible without actually hitting it?

So we've determined that it's extremely close to 0% chance, but is it exactly 0%? Or is it ".000000000000-infiti-0001%"?

:roll: Not again...

 

[DrPizza]

The probability of getting all heads, or all tails is equal to the probability of getting HTHHHHHHHHHHHHHHH.... (continuing with heads forever)
Or, THHTHTHTHTHHHHHHHHHHHHHHHHHHHHHHHHH... (continuing with heads forever)

It's the same as the probability of any infinite sequence of independent flips.

However, this doesn't really make sense, does it?

It's sort of like, "pick a random number between 0 and 1. What's the probability that the number is a rational number?" The probability works out to be zero (there are infinitely more irrational numbers than irrational numbers). Does this mean that there are no rational numbers, simply because the probability is zero? Nope.

Finding the probability of any particular combination is somewhat similar. Let's say that ever time a digit is an even digit (including zero, in case someone doesn't consider 0 to be even), it represents a heads. And, every time a digit is odd, it represents a tails. Thus, the decimal .3232323232... represents THTHTHTHTH...
And, pi - 3 would be the representation of (.14159265...) THTTTHHT...
Of course, in this scheme, there would be a lot of repitition (.232323... is the same as .474547454745... So, we could switch to binary rather than decimal to continue this argument. Regardless, all heads would be .111111111... or .33333... or .111333555... (binary really makes it nicer, since there would only be 1 number which represents each possible coin flip combination.
The probability of any particular order is zero. This does not mean that it doesn't have a representation as a point.

 

[Alonelymuffin]

.0000-infinity-0001>0
It would be something that can only happen once in an infinite number of tries.

I also recall hearing somewhere that in practice, a coin coming up on a given side slightly increases the chances that it will come up on that side again on the next flip.

 

[CycloWizard]

I think instead of saying "0.00000...0001%", people should stick to "infinitesimal".

 

[Smilin]

When you start talking about 'repeating forever' you are dealing with limits approaching zero or infinity. At these extremes you must also start considering the third possibility: The coin could land on edge. :shocked:

 

[Special K]

This reminds me of another "problem" I have seen - suppose you pay $1 million to play a game where you keep flipping a coin until a tail appears. You then win 2^n dollars, where n is the number of consecutive heads you flipped before getting the tail. The expected value of this game is infinite, even though the chances of actually coming out ahead are quite small.

 

[DyslexicHobo]

I dunno if this has been said or not (I'm tired, didn't read all the posts), but I was thinking about this the other day in my lunch period; it came up in a conversation. I realized that the answer is definitely (from what I understand) not equal to zero. If you flip a coin an infinite number of times, it has to land in some sequence of heads and tails. The probability of the coin falling in whichever sequence it actually follows is exactly equal to the probability of the coin landing on heads constantly. So assuming that a coin, in fact CAN be flipped an infinite number of times, the probability of a coin landing on heads an infinite number of times (consecutively) is not zero. This is, however, thinking about it logically. By using the limit process, it's equal to zero. So I like to stick with my psuedo-number: 00000. . .00001.

Bleh, elementary statement. I'm sure this has already been pointed out, but I finally get to have some input!

 

[Who Me]

Originally posted by: Mark R
-------------------
Yes, you are. You are 184,756 times likely to get 10 heads and 10 tails, than 20 tails.

However, it is equally likely to get 10 heads then 10 tails, as 20 tails.
--------------------
Doesn't sound right.
The probability of getting a particular number of sequential heads is surely related to the probability of getting a particular number of heads in a given number of tosses ?

 

[Who Me]

Originally posted by Chcarnage:
----------------
".000000000000-infiti-0001%" is zero. (This one has been around a few times in ATOT)
----------------
Infinity cannot exist within a finite universe.
IOW, you cannot use Inifinity to make real-world calculations.

 

[chcarnage]

Originally posted by: Who Me
Originally posted by: Mark R
-------------------
Yes, you are. You are 184,756 times likely to get 10 heads and 10 tails, than 20 tails.

However, it is equally likely to get 10 heads then 10 tails, as 20 tails.
--------------------
Doesn't sound right.
The probability of getting a particular number of sequential heads is surely related to the probability of getting a particular number of heads in a given number of tosses ?

Originally posted by: Who Me
Originally posted by Chcarnage:
----------------
".000000000000-infiti-0001%" is zero. (This one has been around a few times in ATOT)
----------------
Infinity cannot exist within a finite universe.
IOW, you cannot use Inifinity to make real-world calculations.

You say universe but you meant your intuition. But like everywhere else, intuition can be wrong.

There is only one sequence to flip 20 heads in a row (HHHHHHHHHHHHHHHHHHHH)
On the other hand, there are 184 756 sequences to reach a total of 10 heads and 10 tails.
HHHHHHHHHHTTTTTTTTTT
HHHHHHHHHTTTTTTTTTTH
HHHHHHHHHTHHHHHHHTH
...
Every sequence has the same probability, so "10 heads and 10 tails in no particular order" is 184 756 times more likely to happen than 20 heads "in no particular order, (there is only one)".

Similar to this, the most probable sum of two dice rolls is 7 and not 12 or 2, because there are 6 ways to reach the sum 7 (1+6, 2+5, 3+4, 4+3, 2+5, 1+6) but only one way to reach 12 or 2 (6+6 and 1+1, respectively).

Infinity is both a useful mathematic concept and a reality. Even between 0 and 0.1 there is an infinite amount of irrational numbers. Not a close-to-infinity-really-high amount, but real infinity.

On the other hand, "close-to-X-but-not-really-X-but-really-close", or "000000000000-infiti-0001", like it was called here, is not a mathematically describable concept and not useful.

There are many other elusive but mathematically definable concepts out there. Imaginary numbers (roots of negative numbers) do not "exist" but are of great use for some things. Acceleration is measured in meters/second^2, although no human being ever has witnessed a "square second". Some centuries ago, clerics were sceptic of the number zero and you might have argued that "in a universe that exists, "nothing" can not exist". Why, of all these things, it's infinity that still is disputed, I do not know.

 

[Who Me]

Ask yourself what Infinity is.
It's non-finiteness.

What is finiteness ?
It's dimensionality.

What is dimensionality ?
It's that which can be measured - or measurability.

And to be measurable, something must be finite.

Non-dimensionality (ie - undividedness) cannot exist within a finite (ie - dimensional, divided) universe.

They are mutually exclusive.
As soon as you introduce dimensionality into Infinity, you destroy it - because you've divided it.

And as soon as you introduce Infinity into dimensionality, you destroy it - because to render something infinite is to remove all division.

 

[CycloWizard]

I think the problem of 'infinity' is a purely mathematical one that arises out of certain simplifications leading to singularities or as a result of certain simplifying assumptions, such as a far-field condition.

For example, in the problem of elasticity, stress is infinite at a sharp corner (because stress is the force divided by the area of application of said force). However, in reality, there is no such thing as a 'sharp' corner, as all things have a finite dimension (i.e. even if it is only the size of a single molecule, the corner still has some characteristic curvature). Thus, the force is really being applied always to a finite area and the stress will always be finite. The one case where this hasn't been settled (as far as I know) is in the case of an impact. When two bodies meet at t=0, the stresses inflicted on both bodies are infinite. I'm not too familiar with this area though, so a similar argument may apply.

Similarly, if we consider the problem of *insert any type of field equation here* in a semi-infinite medium, 'semi-infinite' is simply a hand-waving way of mathematically declaring a far-field condition. If I lie down on the earth, the heat from my body will only go so far before it has negligible effect on the temperature distribution within the earth. Therefore, it is safe to say that the earth is a semi-infinite medium in this case.

 

[RossGr]

Originally posted by: HeartView

Originally posted by: mozirry
I think it kind of works like this.


One flip 50% chance of getting heads

two flips 33% chance of getting both two flips heads
(Possible choices, (1st flip/2nd flip) heads/heads, heads/tails, tails/heads, tails/tails))

three flips 16.66% chance of getting all three flips heads
(possible choices (1st flip/2nd flip/3rd flip) (heads/heads/heads, heads/tails/heads, heads/heads/tails, tails/heads/heads, tails/tails/heads, tails/tails/tails)

Except that ALL combinations of heads and tails have the same chance of happening assuming the toss happens under the same circumstances each time. So you are no more likely to get 10 heads and 10 tails in 20 tosses than you are of getting all 20 heads or all 20 tails. You have the same chance each time of getting heads or tails, regardless of what came up in previous tosses.

this is just wrong. There is exacty 1 way to get 20 heads, there litterly thousands of differnent ways to get 10 heads and 10 tails. Now you are correct that if you specify the exact order that the 10 heads and 10 tails show up you have the same odds as getting 20 heads. But that is a bit different then just looking at the odds of a final result of 10 each.

edit: a more realistic guestimate at the number of combinations possible.

 

[Who Me]

Post-script on Infinity.

I'm not sure it gets you anywhere to think about what is 'at the edge' of the Universe (say you could be at the leading edge of the explosion that was the Big Bang).

We cannot escape dimensionality (the Universe).
We cannot exist outside it.

Thinking of the Universe as a 'big space' (something that can be moved outside of) serves no purpose, because to regard it purely as a spacial entity is to ignore what it really is ... space-time.
The two are inextricably interwoven, they are one - one cannot exist without the other.

So you can't talk about escaping space.
You have to talk about escaping space-time.

And we simply cannot move outside space-time.

 

[chcarnage]

Infinity exists. P.e. it's the amount of irrational numbers between 0 and 0.1, like I said.

And even if nobody ever will toss a coin for an infinite amount, you still can calculate the probability of a certain sequence. Mathematicians can navigate in a space with five spatial dimensions no matter if this concept is a reality or not.

 

[CycloWizard]

Originally posted by: chcarnage
Infinity exists. P.e. it's the amount of irrational numbers between 0 and 0.1, like I said.

And even if nobody ever will toss a coin for an infinite amount, you still can calculate the probability of a certain sequence. Mathematicians can navigate in a space with five spatial dimensions no matter if this concept is a reality or not.

Are irrational numbers real or are they simply a mathematical abstraction? Something existing in a mathematical sense is not necessarily the same as something existing in the real world.

 

[smack Down]

Originally posted by: CycloWizard

Originally posted by: chcarnage
Infinity exists. P.e. it's the amount of irrational numbers between 0 and 0.1, like I said.

And even if nobody ever will toss a coin for an infinite amount, you still can calculate the probability of a certain sequence. Mathematicians can navigate in a space with five spatial dimensions no matter if this concept is a reality or not.

Are irrational numbers real or are they simply a mathematical abstraction? Something existing in a mathematical sense is not necessarily the same as something existing in the real world.

Sure just make a right triangle where the two small sides have a length of 1. The third side with have a length of sqrt(2).

 

[Who Me]

Infinity is not an EXTENSION of real-world.
Just as division by zero is mathematically impermissible, so to is multiplication by Infinity (same 'calculation').
The Universe is composed of a finite number of particles.
Which means that there is a maximum real-world number.
This is the total number of divisions (permitted by the laws of physics) of that which is physically divisible (ie - of matter).
'Division' of that which is PHYSICALLY indivisible (eg - space and time) comes under the heading of mathematical abstraction - because it cannot be applied to the real, physical world.
And since there is a limit to the total number of particles and their physical divisibility, there is also a maximum real-world number.
Ergo, Inifinity does not exist in the real world and cannot be applied to real-world calculations.

 

[gbuskirk]

I say the answer is (1-0.999999...). Which is indistinguishable from 0.

Who Me, you have a lot of confidence that there is one single indivisible fundamental particle. I don't know if that's a prevailing scientific opinion. What is the nature of matter?

 

[Who Me]

An interesting point, GBU.
I think 'real-world' applies to the Newtonian rather than the quantum domain.
The real world extends as far as the point at which the division of a particle still produces particles of matter with all the attendant properties thereof ... but no further.

 

[zerogear]

IMO there is no 'randomness' in flipping a coin. Its simply that we haven't been able to determine all the actual factors, when we can't calculate every single factors of the chance, it becomes 'random'