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Three insects are going in a row which is just like a line segment.

I ask the first insect,"How many insects are behind you?"

He answered, "Two".

I ask the second insect,"How many insects are behind you?"

He answered, "Two".

I ask the third insect,"How many insects are behind you?"

He answered, "Two".

Great Mathematicians

How is this Possible?

coody wrote:

In a gunball machine, there are 2 orange gunballs for every 3 yellow gumballs. there are 80 orange gumballs in the machine. find the number of yellow gunballs in the machine.

You are given the proportion of orange gumballs to yellow gumballs. In other words, you know that for every 2 orange gumballs, there are three yellow gumballs. You also know that there are 80 orange gumballs. In math terms, this means:

where *x* is the number of yellow gumballs in the machine. So, solve for *x* to answer the problem.

bobbym wrote:

Secondly, I am not a big fan of it. Like Einstein I do not like anything about quantum mechanics.

Many arguments with my brother on Deutsch's many universes and quantum computing made me a detractor.

When a particularly annoying tech type said to me that quantum mechanics was definitely true beacuse when they a do a calculation using it they get 8 digits of precision I wished I was in another universe. One without people like him. Sorry.

Always remember:

All models are wrong. Some are useful.

bobbym wrote:

Hi euclidcries;

Non singular and invertible are the same. They mean a square matrix that has an inverse.

An equivalent definition is a square matrix with a non-zero determinant.

euclidcries wrote:

Also, how is this done for infinite sets?

Just because two sets are infinite does not imply that their cardinality is the same. In other words, given two infinite sets |A| and |B|, we cannot conclude that |A|=|B| (although sometimes this is true).

For example,

**E** = {x | x/2 ∈ ℤ} is the even integers.

let f:**E**⇾ℤ defined by f(x)=x/2. f is a bijective function, so

|**E**|=|ℤ|.

But, can you come up with a bijection between ℤ and ℝ?

euclidcries wrote:

when you say "injective function" are you saying one-to-one correspondence?

No, not quite. An injective function is another name for a one to one function, which implies the following:

if f(a) = f(b), then a = b.

A one to one correspondence is another name for bijection between sets.

but, what does the notation |A| or |B| mean? (when you wrote |A| = |B|, what is this statement saying?) Also, how is this done for infinite sets?

|A| is the cardinality of set A.

By definition, |A| ≤ |B| if there is a one to one mapping of A into B. By the Schröder-Bernstein Theorem, if |A|≤|B| and |B|≤|A|, then |A|=|B|, which implies that there is a bijection between A and B. (See: <http://plato.stanford.edu/entries/set-theory/primer.html#5> and <http://mathworld.wolfram.com/Schroeder-BernsteinTheorem.html>)

I'll try to address the rest of your post later. I have to get to class right now.

Are you wanting to show that two sets have the same cardinality?

SuperLynx wrote:

What is this

Rprogram you speak of ???:)

From the Wikipedia article (*bold face emphasis mine*)

http://en.wikipedia.org/wiki/R_Statistics :

R is a programming language and software environment for statistical computing and graphics. The R language has become a

de factostandard among statisticians for the development of statistical software, and is widely used for statistical software development and data analysis.

Although R is mostly used by statisticians and other practitioners requiring an environment for statistical computation and software development,it can also be used as a general matrix calculation toolbox with performance benchmarks comparable to GNU Octave or MATLAB.

Statistics relies heavily on matrices, because they can greatly simplify things. For example, when performing linear regression analysis, many of the matrix form equations are identical, whether one predictor variable is used, or 10,000 predictor variables are used. R handles matrices extremely well and simply. For example, obtaining the determinant of a 100×100 matrix (or even much larger matrices) named X is as simple as typing "det(X)".

Note that R is for numerical analysis and is not a computer algebra system.

R is very powerful, and *free* (compare to S-Plus, at $2399 per year or to SAS, at $6000 per seat). It is available for Windows, Mac OS X, and Linux in binary form (i.e., just download and install), and the source code is available for compiling on other platforms.

To learn more and to download R, see:

http://cran.r-project.org/

GeniusIsBack wrote:

( THE ROOT IS WHAT YOU THINK! Infinite-Recurring 0.9 Is )

I just Gave this to a Goldfish! and it gave a Better Answer than what I have seen So Far!

GiB.

I think, and one of the definitions you provided stated, that ROOT is simply an informal way of shouting "square root of." Thus, ROOT 100 = √(100) = 10.

If you are using ROOT in a different manner, please provide a *rigorous* definition. If you don't explain your non-standard terminology to your audience, then, however genius your claim may be, it is indistinguishable from gibberish.

GeniusIsBack wrote:

This is Incredible!!

( THE ROOT IS WHAT YOU THINK! Infinite-Recurring 0.9 Is )

The Same as ( THE ROOT For 100 is 100 )

GiB.

How is it incredible? Do you find it strange that, upon introducing new mathematical terminology to mathematicians, that those mathematicians expect that terminology to be rigorously defined before using it?

GeniusIsBack wrote:

Just For ALL_Is_Number

100 Root = 100 ( THE ROOT IS THE START! THE ORIGIN! ) (The Root) is Not The Square Root) And Just In case Anyone Thinks It's Teeth? Etc.

GiB.

That isn't a definition.

GeniusIsBack wrote:

Quote: ALL_Is_Number (Strange Name!) "Any confusion is due solely to the unnecessary new terminology, not due to the fact that 0.999999 is exactly equal to 1. Please define the terminology you are using. "

GiB. The Terminology is Quite simple!...

(d) (Math.) That factor of a quantity which when

multiplied into itself will produce that quantity;

thus, 3 is a root of 9, because 3 multiplied into

itself produces 9; 3 is the cube root of 27.

Root of an equation that value which, substituted

for the unknown quantity in an equation, satisfies the

equation.

I've culled the only two mathematics related definitions of *root* from your post. You've explicitly stated it is not the first, and the second doesn't fit in the context you've presented. S, again, please post a *mathematical* definition for *root* as you are attempting to use it.

D'oh! I forgot all about integrating over the normal curve! That would have been easy enough to do.

bobbym wrote:

Hi All_Is_Number;

Excellent that is correct!

P(having a losing day ) = 0.0001212752342855117

3.677416458992032 standard deviations

He has approximately 1 chance in 8245 of having a losing day.

How did you obtain such precision with P(losing day)?

bobbym wrote:

What can I say to the casino manager about his chances of losing on any single day?

Assuming

games are played each day, the probability of the casino losing money on a particular day is

bobbym wrote:

Yes, you are paid even money.

bobbym wrote:

Hi All_Is_Number;

I will change the problem to make it less ambiguous.

3) A guy owns a small casino. It only has one gaming table. He would like to earn 1000 dollars a day with it. There is only a flat bet of 2 dollars per roll and the players probability is 18 / 37 of winning that bet. How many games must be played on average for him to earn his 1000 dollars. Since he doesn't like to lose he needs to also know what is his chance of losing in any single day.

That clears up everything. I will adjust the original post when the smoke clears.

Not to be nit-picky, but what is the payout of the bet? Two dollars? A flat bet of $2 implies only that every bet costs $2 to place. It does not imply a particular amount paid out if the bet is won.

bobbym wrote:

Hi All_Is_Number;

I am working on the solution now as I cannot find it. But please work on your answer as it is not correct.

If, by "players expectation is 18 / 37 dollars per game" you mean expected value of each bet, and not expected gain on each bet, my answer is correct.

If you meant expected gain, my methodology is incorrect. If you meant the probability of a player winning a particular bet is 18/37, then my methodology is incorrect.

bobbym wrote:

Hi All_Is_Number;

Thanks for looking at the problem but there is definitely if I remember an answer for every part of it. The problem answer is not posted here because It was being contested on another forum ( now defunct ). Since they were aware of me here I did not want them to just click a hide button.

I'm reasonably confident more information is needed. In particular, I believe the number of bets actually laid, and the probability of a success for each particular bet offered (information not available from the expected value alone) is needed.

I would be interested to see your solution, including how you worked it out. I've been wrong before, and the probability of me being wrong again at least once at some point in time is 0.999999 . :-D

bobbym wrote:

3) A guy owns a small casino. It only has one gaming table. He would like to earn 1000 dollars a day with it. There is only a flat bet of 2 dollars per roll and the players expectation is 18 / 37 dollars per game. How many games must be played on average for him to earn his 1000 dollars. Since he doesn't like to lose he needs to also know what is his chance of losing in any single day.

I didn't see an answer posted for this question.

The problem lacks sufficient information to determine the casino's probability of loss on any single day. As a puzzle for the puzzle maker, I'll let you figure out why! :-) But, given the law of large numbers and the incredible expected winnings of the house, the probability will undoubtedly be VERY low.

GeniusIsBack wrote:

Before any Confusion!? ( The Root ) is Not The Square Root...

GiB.

Any confusion is due solely to the unnecessary new terminology, not due to the fact that 0.999999 is exactly equal to 1.

Please define the terminology you are using.

0.9999999 = 1 (exactly, not approximately). This is easily provable.

Thus, -0.9999999
= -1,

√(0.9999999
) = √(1) = 1, and

+0.9999999
= +1 = 1.

I don't understand why it would be necessary to have a new way of looking at it.

Just in case someone does a search and finds this thread, the solution for the example problem from post 1:

Prove or provide a counterexample:

Let

f:A⇾B. IfC⊂A. ThenC⊂f⁻¹(f(C)).

Proof:

Let *f* : *A* ⇾ *B* = *f* (*a*) = *a*², *C* ⊂ *A*, and *x* ∈ *C*.

*f* (*x*) ∈ *f* (*C*) and *f* (*C*) ⊂ *B*.

*f* ⁻¹(*f* (*C*)) = {*α* | *α*² ∈ *f* (*C*)}.

*f* (*x*) ∈ *f* (*C*), so *x* ∈ *f* ⁻¹(*f* (*C*)).

∴ *C* ⊂ *f* ⁻¹(*f* (*C*)). ∎