Math Is Fun Forum

Always remember:

All models are wrong. Some are useful.

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?

If there exists an injective function from A to B and there exists an injective function from B to A, then |A| = |B|, and therefore there is a bijection between A and B.

From the Wikipedia article (bold face emphasis mine)
http://en.wikipedia.org/wiki/R_Statistics :

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/

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?

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.

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

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.

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.

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.

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

Proof:

Let f : A ⇾ B = f (a) = a², C ⊂ A, and x ∈ C.

f (x) ∈ f (C) and f (C) ⊂ B.

f ⁻¹(f (C)) = {α | α² ∈ f (C)}.

f (x) ∈ f (C), so x ∈ f ⁻¹(f (C)).

∴ C ⊂ f ⁻¹(f (C)). ∎