Math Is Fun Forum

mrpace

Member

Registered: 2012-08-16

Posts: 88

Probability problem

Imagine there is a field of infinite daisy chains.

The probability that a given chain contains n daisies is given by the following.

P = 2 / (3^n)

What is the average length of the daisy chains in the field?

Thank you.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: Probability problem

hi mrpace

Imagine there is a field of infinite daisy chains.

The probability that a given chain contains n daisies is given by the following.

P = 2 / (3^n)

What is the average length of the daisy chains in the field?

Is this the exact wording of the question?  If the field has infinite daisy chains, to me that means every chain is infinite in length.  So I guess that isn't what is intended.

If there are an infinite number of chains, each of which has some finite number of daisies, then the answer cannot be determined.  Say nearly every chain has 3 daisies and just a tiny number have a different number of daisies then the expected length would be close to 3.  Substitute x for 3 and you can see that any answer is possible because we don't know how many chains there are of each length.

The only way I can make sense of this is if the question reads "A field has an infinite number of daisy chains.  One chain has length 1, one chain has length 2, one chain has length 3 and so on.   This I can work out and I get the expected length = 1.5

If this interpretation is what you want then post back and I'll complete the proof.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Grantingriver

Member

Registered: 2016-02-01

Posts: 129

Re: Probability problem

Hi bob bundy, in fact, your reasoning is correct, since the problem is a weighted arithmetic mean and each probability is actually the weight of the corresponding team (the total of the weights of all terms is 1). You don't have to derive a closed formula of the sum (I don't try, but from my experience, in these types of expressions, there is usually no closed form). However, we have:

therefore, the series converges and hence it has a sum, but if we notice that:

and the remaining terms will be continuously lesser and lesser than that, so the convergence is fast and then we can calculate the average to the accuracy of the second decimal place by simply adding the first seven terms:

Grantingriver

Member

Registered: 2016-02-01

Posts: 129

Re: Probability problem

Some elaborations to whom it may interest. The number "n" in the preceding argument is a discrete variable hence, by definition, we can not use the differentiation to calculate its limit (by applying L'Hopital's Rule). However, if we define a continuous variable x on the interval [1,ꝏ] , then we can write x=n+m where n and m are the integer part and the decimal part of x, respectively. So we have:

but we also have:

now from the first result we get:

but for any value of x (whatsoever large) there is a finite number m such that:

so we have:

and since both terms in the last equation are positive, we have:

Therefore the conclusion in the previous argument is correct. It follows that treating the discrete variable as a continuous variable (in the preceding problem) is a valid assumption. However, the simple and more appropriate argument in this case goes as follows:

and hence:

Last edited by Grantingriver (2018-05-15 20:32:05)

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: Probability problem

hi Grantingriver

The sum can be written as a series of GPs.  Simplifying, it becomes a single GP with a sum of 1.5

If GP(a,r) means a GP with first term a and ratio r:

If |r| < 1 then sum to infinity is a/(1-r)

 

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Grantingriver

Member

Registered: 2016-02-01

Posts: 129

Re: Probability problem

Excellent!!

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: Probability problem

Thanks!  I think I deserve an award for getting all those brackets right! 

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Grantingriver

Member

Registered: 2016-02-01

Posts: 129

Re: Probability problem

Of course!! And you deserve an addisional award for your creativity also

mrpace

Member

Registered: 2012-08-16

Posts: 88

Re: Probability problem

Thanks for the responses guys.
I actually figured it out before checking back here, which pleases me greatly.
Yes the correct answer is 1.5 and no the question was not out of a book or anything, I just wondered it myself.