Math Is Fun Forum

But by what means? The technique to efficiently replace the carbon energy hasn't been invented.:(

Since no one seems interested, I have to solve it myself:

n balls;
b bins;
k times independent and different toss and one repeated toss before game over.

Now the issue important is the probability of the event that the initial k times of toss are into one-by-one different bins, among which is the destination of the k+1th ball.

The method to solve this probability, now defined as P(k), where 1≤k≤b-1,(n≤b is defaulted) is rather macro, more specifically, the Classical Probability Model. First, toss all the n balls, observe each ball's destination, so  number of all the possible exclusive sequences of the destinations of 1st to nth balls is b^[sup]n[/sup]. For each ball has b destinations. Moreover, these sequences not only share the same probability, they altogether are all the possibility as well. Hence, the probability of each sequences is 1/b^[sup]n[/sup].

Now we need to count how many ways there are to satisfy the out come of k different destinations, then one repeated destination, and then, if any, n-k-1 arbitary destinations. Apparently the number of ways is:

So the expected number of all k's is

Then who can simplify the formula above out??????????????????

Yes, the interaction between ocean temparature and the Arctic Ice.

We may also note that the temperature rises in an expontial manner as well...

Mmmm, I share the same problem with Mikau. My long term memory is not good, either. Actually the partial reason why I love maths is that it is the subject that requires the least memorizing!! Literature is always the last subject I want...

differentiate the second equation by t on both sides:

substitute dT/dt by the equivalent in terms of LI and t in the first equation, we can get

Note two  functions equal to their 2nd derivative respectively, which are:
C sinh(t) and C cosh(t)

It is okay to conclude the function of Lack of Ice in terms of t shall be:

By knowing the initial condition of LIo and To( hence LI'o), we can determine the function.

Both Sinh(t) and Cosh(t) are more or less expontial

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I find it convincing for this example:
y=1.0001^n+some random variable with a variation of 0.001
draw y.
At early phase, constant model is best in explaining the graph.
then from some point, linear model is better.
At the end, you may realise it is in fact expontial, but it is toooooo late.

Hard question!
But I got it!
After reviewing the Classic Probability Model, anyone could solve this problem!

Hint: First consider all the possiblities to throw all balls one by one. The number of combinations is
binnumber^ballnumber. Apparently each possiblity have the SAME chance. Then all you have to do is to count how many ways to arrange k different bins+the bin repeating the kth+ the rest random bins.

Anyone like to solve it thorough??:P:P

Why is this?  How do you know a measurement, any measurement can be described in a finite amount of digits?  How do you know that it isn't just an artifact of our measurement technique which produces finite digits?

Even if the thing we are measuring is continuous (height, for example), because of our limited technology, we may only measure discrete distances.  You seem to be claiming you can accurately measure reality with no error.  Certainly, that is not the case.
---I mean matters-wood, plastics, paper, to name a few. They are made of an integer amount of particles, disabling them to form any real circles.

First off, show me how my definition is not logically consistent or does not mirror reality.  Actually, first, you can show me where infinity occurs in reality.  Or do you think infinity should not be part of mathematics too?  But I believe definitions are still arbitrary.  It is arbitrary is the same way that the word "chair" is arbitrary.

--- I mean 0.999... with infinite digits is not, at least the form. So with a line formed by points.Secondly, I need not show you where is infinity. Instead, you need to show me how can a line be formed by "infinite" points in reality.
The English word "chair" may be arbitary, but the concept of "chair" is based on a repetitive object designed to  hold an math with a surface and usually 4 legs beneath the surface.

I'm back

"Most people accept this definition by intuition alone.  You're the first person I've ever seen which did not."-Ricky
Since you have explained this sentence enough,  I will no longer indicate that you were trying to judge me as "absurd".

I may also note you a phsycologic fact that people tend to stick to what they were told during their childhood with prejudices without enough questions, making it difficult for a heratic opionion to spread. For example, after my post exploring the inconsistency of infinity, still a lot of people tried to amend the "infinity" concept rather than to throw it away. Also some one sticked to 1-0.999...=0.00..1 where 0.00...1 is nothing or invalid, while in fact 0.00..9 is invalid according to the same logic.

On the popularity issue, I shall argue that infinite expansions never rivals sufficient but finite expansions when the real thing is clear. Pi is great, but it is more accurate to say that all "circles" we can encounter coincide with Only a finite digits of Pi. So my alternative to replace an infinite expansion is just sufficient expansions, or their group. Actually is my alternative so far that all the people of the world along with all the computers of the world can practice?

At last, a defination cannot be ARBITORY, it should be logicly consistent with itself and should mirror the reality. And these are far more important than intuition alone-people used to have many intuitions, including one that the sun rotates around the earth.

To Anthony:
You cannot agree that 1/3=0.333... for it is easy to use the "proof" with multiplying them with 3.
Instead, you should point out no one has calculated 1/3 out yet.(As I insisted in the previous thread, which has already been closed by Ricky)
Or simply further, 0.333... is inconsistent with itself. (As I insisted in this thread)

Perhaps because Ricky believes what he has leant from books or what he has been taught are correct.

x = 1/2y + 1/2
-in my book this is equally an inversefunction.
Swap x and y, however, doesn't change the nature of a function. In the function x = 1/2y + 1/2, the dependent variable is one half of the inverse of the independent variable plus one half.(dependent variable is the one on the left of the equater) In the function y = 1/2x + 1/2, the same thing stands. Since they could be called the same functions by nature, it is at least of some reason to say they are both the inverse functions.

Further, naming y = 1/2x + 1/2 the inverse function of y = 2x - 1 enjoys pratical benefits. Almost all the functions have the form y= some x, therefore we can merely figure out two inverse functions among the mass functions we encountered unless we define that way. Inverse functions defined in this way are symmantric to the line y=x. Hence in this way, we actually reduce the functions we need to learn by half, in the field of how to graph, for example.

In general, both ways are right, and the latter helps us a lot!