Mathematical Probability Question

[Legendre]

I spent quite a lot of time trying to figure out this question that I read in a book. Through trial and error, I got a rough idea of what the solution should be like but can't figure out how I would derive it mathematically. Can anyone give me some directions? :)

QUESTION:

n people stand in a circle, where n > 2. we pick two person, A and B, at random. what is the probability that there are exactly r people standing in between A and B, in the clockwise direction?

 

[tiny-tim]

Hi Legendre!

Can you think of a way of simplifying the question?

 

[elibj123]

Given a choice of A, how many B's will satisfy the condition?

 

[tiny-tim]

Given a choice of A, how many B's will satisfy the condition?

Yup!

(or, rather, what proportion of B's will satisfy the condition?)

So you can simplify the problem by arbitrarily fixing A, which makes the answer … ?

 

[Redbelly98]

Moderator's note: thread moved to Homework forums.

 

[Legendre]

Yup!

(or, rather, what proportion of B's will satisfy the condition?)

So you can simplify the problem by arbitrarily fixing A, which makes the answer … ?

Thanks for not outright giving the answer! I find that I learn much more if I were to come out with an answer using hints.

Now let's see...

Fix A. Counting in the clockwise direction, only 1 B satisfies "exactly r people are standing between A and B".

0 <= r <= (n-2). So there are (n-1) possible r.

So the probability is 1/(n-1).

 

[tiny-tim]

(just got up :zzz: …)

So the probability is 1/(n-1).

Woohoo! ​

 

[Legendre]

(just got up :zzz: …)

Woohoo! ​

haha thanks!