Let's discuss Mathematics

[Mise]

Guys, is there an easy proof/simplification for the following thing:

If you have a 1 in x chance of something happening, then after x attempts, the probability of that thing not having happened yet is p = (1-1/x)^x. Now, as x->infinity, the probability tends to 1/e (I know this because I recognise the numbers when I calculate it). But how do you prove that? Do you have to do some sort of clever series thing to expand the bracket when raised to the power of x?

The rule of thumb is useful in board games: if I select a thing that has a 1 in 20 chance of happening during each turn, then I know that, after 20 turns, there is a ~2/3 chance of it happening at least once. In Settlers of Catan, for instance, it's easy to calculate the probability of rolling a 4 as a 1 in 12 chance (in fact, the number of dots on the hex tells you that it's a 3 in 36 chance). So you can estimate that, with 4 people playing, there'll be a 2/3 chance that a 4 will be rolled at some point during the next 3 rounds (i.e. after the dice have been rolled 12 times). Obviously for small x it doesn't work so well, but for small x it's easy enough to do in your head.

 

[Petek]

The exponential function e^x satisfies the equation

e^x = lim (n--> infinity) (1 + x/n)ⁿ

(https://en.wikipedia.org/wiki/Characterizations_of_the_exponential_function)

Let x = -1 to get the result you seek.

 

[Mise]

Ahh okay, I'm not familiar with that one! Makes sense. Thanks!

EDIT: The "equivalence of 1 and _" sections of the wiki article are especially useful. I couldn't really "intuit" #1 initially, but the equivalence proofs helped me make sense of it.

 

[Robert FIN]

A short question.

Can you divide by cos (or sin) in an equation? I'm just worried if it is zero and then it would be wrong. Do I have to test it afterwards in case it was 0?

Thanks. I hope the question makes sense.

EDIT: I meant "cos x" or "sin x" ofc..

 

[dutchfire]

cos(x) * A = cos(x)*B means that either A=B or cos(x)=0.

 

[Atticus]

Can you divide by cos (or sin) in an equation? I'm just worried if it is zero and then it would be wrong. Do I have to test it afterwards in case it was 0?

You can and you have to.

If you have some specific problem, don't hesitate to ask.

 

[azzaman333]

A short question.

Can you divide by cos (or sin) in an equation? I'm just worried if it is zero and then it would be wrong. Do I have to test it afterwards in case it was 0?

Thanks. I hope the question makes sense.

EDIT: I meant "cos x" or "sin x" ofc..

I'm pretty sure by definition tan(x) = sin(x)/cos(x), so yes, it's definitely possible.

 

[Robert Can't]

I'm pretty sure by definition tan(x) = sin(x)/cos(x), so yes, it's definitely possible.

Possible for all x not equal to (pi)(2n-1)/2.
The function is not defined for those values.

 

[Robert FIN]

Are you prepared for another one?

Let's say f(x) = x tan2x . Then we count f'(x). I think it is called derivation in English.

So the counting goes right when I do this:

tan2x * 2 + x(1+tan2x^2) * 2

But that seems weird. I forgot why did I put the first * 2 and I don't know why the last * 2 is there neither.. I know that it has something to do with the inner function but then is it required in the first * 2 and I cannot see how the second * 2 comes from the inner function. I totally lost the idea..

 

[Lohrenswald]

derive sin(x)/cos(x) instead

you know the formula for derivating fractions right?

 

[Robert FIN]

You mean converting tan2x to sin2x/cos2x ? That might work but it feels kinda tricky...

 

[Lohrenswald]

It's probably easier than trying to fiddle with cotangens
Fraction derivation require less insight to do I think.

If I have time and resolve later I'll post how to do it.

 

[Robert FIN]

Alright, thanks.

So the thing is that you have to count "D" for the outer function and then you count that times the "D" of the inner function. But the weird thing in my counting was that why do we need to count the "D" of inner function even though we didn't have to count it for the outer function...

 

[Lohrenswald]

I read your first post more closely
There are two problems:
You have forgotten the multiplication rule (though you use it correctly there) and
You derivated tanx wrong
Again, I'll show you later though

 

[Robert FIN]

I read your first post more closely
There are two problems:
You have forgotten the multiplication rule (though you use it correctly there) and
You derivated tanx wrong
Again, I'll show you later though

Okay, this is true. I think I remember the multiplication rule now somehow but the first part of the calculation is to derivate x which is 1 and then it is 1 times the tan part. So basicly in the first operation we didn't have to derivate the tan part but the counting goes right if I put the D of the inner function there even though we didn't derivate the outer function. Is that how it is supposed to work?

Also I cannot see how I derivated the tan2x wrong.

And I don't know what you mean by later but I have to add that this is for practicing to a math test which is tomorrow so it would help a lot more if you could check it out today. Helping is voluntary of course but I still wanted to tell this fact

 

[Lohrenswald]

Okay, this is true. I think I remember the multiplication rule now somehow but the first part of the calculation is to derivate x which is 1 and then it is 1 times the tan part. So basicly in the first operation we didn't have to derivate the tan part but the counting goes right if I put the D of the inner function there even though we didn't derivate the outer function. Is that how it is supposed to work?

No

You're not supposed to derivate the inner function if the outer isn't.

Derivative of tanx is 1/(cosx)^2

I'll show you the whole thing in I hope within 2 hours. Maybe it'll be 3 but I'll try to avoid that

 

[Robert FIN]

No

You're not supposed to derivate the inner function if the outer isn't.

Derivative of tanx is 1/(cosx)^2

I'll show you the whole thing in I hope within 2 hours. Maybe it'll be 3 but I'll try to avoid that

Thanks for the help.

I don't know why the counting goes right if I do the wrong thing (this is just a part of one big counting though). Maybe we'll find out in a few hours

However, I am quite sure that tan(x) can be also derivated to 1+tan(x)^2 . I have used it successfully in other calculations...

 

[Lohrenswald]

It's not that you do the wrong thing, it's that you doubt yourself

But a mistake on my part, 1/(cosx)^2 ans 1+(tanx)^2 are identities, ,so they're both right.

But I on my way home now, so I'm a little over an hour (I promise) I'll show you why the first one is the more obvious solution (basically I'll write it on a piece of paper and scan it )

 

[Lohrenswald]

I shure hope this is understandable

Spoiler :

 

[Robert FIN]

Thanks a lot. Now I will try the calculation again using those tips. I got the main points but your writing is quite... special .

I understand now that it was all good exept that I added the ' of inner function when it was not needed.

I will post you back here when I have tried that. Thank you really much, you are really helpful. I can't believe how someone can help a weirdo like me, lol