How can we inpretate for falling many sphere?

[Choi Si Youn]

Hi, in this thread I want to discuss some problem that I wonder.

I'm interested in "Probablilty"

The first wondering is How can I express that probability follow an exponential.
I'm wonder why probability expess in exponential?
and Can I show it, if I experiment something, that graph is exponential.

So I am setting some kind of experiment.
Express that experiment, I use speaker that I used for audio.
and I use function generator and audio amp.


I give a signal from function generator to speaker with surronded by sylindrical plastic cover.

and I put a BB(plastic sphere) into a speaker.


then I turn on the audio and function generator and check the height of jump of BB.


is it ok for my purpose?

and I still don't know that why probability follow the exponential?

and Why that BB jump randomly? the function generator and the audio give a same energy.



I can't apply between the textbook when I studied and the actual.

 

[kuruman]

Before you design an experiment, you need a theoretical background to guide your experimental procedure. Here, I assume you measured the height of the jumping BB and that's your dependent variable. However, you do not mention what your independent variable was. What did you vary in a controlled way as you measured the height? You mentioned a graph. Did you make one? If so, what was on the x-axis? Besides all this, what reason do you have to believe that the height of the BB has an exponential dependence on your independent variable, whatever that is?

The hallmark of the exponential dependence of a dependent variable $y$ on an independent variable $x$ is that the rate of change of $y$ with respect to $x$ is proportional to $y$. Mathematically, this is expressed as
$$\frac{dy}{dx}=ky.$$Here, $k$ is the constant of proportionality. Before you design an experiment to test the exponential dependence, you need to have a reason to believe that your dependent and independent variables more or less obey this relation. For example, a well known and easy to do exponential experiment is based on coin-tossing. Here's what you do

Start with a large number of identical coins, say 100.
Place them all "heads" up. The number of heads is your dependent variable $y$ and the number of tosses is your independent variable $x$. Your first data point {x,y} is {0, 100}. Now toss the coins up in the air. Say you get 52 heads and 48 tails. Your second data point is {1,52}. Remove the 48 tails and toss the remaining 52 heads. Your third data point could be {2,23}. Remove the coins landing as tails and so on until there are no heads left. Plot number of heads vs. number of tosses; the plot should be an exponential.

Why do I think it should be an exponential? Because I believe that there is a 50-50 probability that a coin will land heads or tails. This means that the number of coins that land heads per toss ($dy/dx$) is (more or less) half the number of the heads that I am tossing ($k=1/2$). See how it works?