# Spin a dial that has a pointer to n regions

• lizarton
In summary, there are a total of n^3 possible outcomes when spinning a dial with n region numbered 1,2,...,n.
lizarton

## Homework Statement

Consider a dial having a pointer that is equally likely to point to each of $n$ region numbered $1,2,...,n.$ When we spin the dial three times, what is the probability that the sum of the selected numbers is $n$?

## Homework Equations

A Theorem I believe is relevant:
With repetition allowed, there are $\left(\stackrel{n+k-1}{k - 1}\right)$ ways to select $n$ objects from $k$ types. This also equals the number of nonnegative integer solutions to $x_{1} + \ldots + x_{k} = n.$

My problem is identifying $n$ and $k$ in these problems. Any help would be greatly appreciated!

## The Attempt at a Solution

I have to use summations, and I'm sure binomial coefficients. I believe that this is a selection; it seems to imitate rolling an $n$-sided die three times, but I even have trouble computing that problem.

The total number of outcomes is $n^3$ (I think)

I started counting ordered triples of some n terms...

$n=3:$ There is only one way, ${(1,1,1)}$
$n=4:$ There are 3 ways, ${(1,2,1),(1,1,2),(2,1,1)}$
$n=5:$ There are 6 ways, ${(1,1,3),(1,2,2),(1,3,1),(2,1,2,),(2,2,1),(3,1,1)}$
$n=6:$ There are 10 ways...
$n=7:$ 15 ways...

For arbitrary n, you can start making ordered triples...
$(1,1,n-2) (1,2,n-3) (1,3,n-4) \ldots (1,n-2,1) (1,n-3,2) \ldots$

Simplify to two spins and draw a table. Then think how that generalises to three spins.

Alternatively look into how to determine the PDF of a sum of independent random variables from their PDFs. This is the more powerful way of doing it because it works for any PDF, discreet or continuous. But the first way will solve the problem at hand.

I'm not even sure that I know how to form the simpler problem, but here's a shot:

For two spins, there are $n-1$ combinations that will sum to $n. (1,n-1) (2,n-2) ... (n-2,2) (n-1,1)$

The total number of outcomes is n^2 (I think?).

Where do I go from here?

## 1. What is the purpose of spinning a dial with a pointer to n regions?

The purpose of spinning a dial with a pointer to n regions is to randomly select one of the n regions. This can be used in various applications such as games, experiments, or randomizing choices.

## 2. How does the dial with a pointer to n regions work?

The dial with a pointer to n regions works by having the pointer attached to a central axis that can rotate. Each region on the dial is marked with a number or symbol, and when the dial is spun, the pointer will land on one of the regions at random.

## 3. Can the number of regions on the dial be changed?

Yes, the number of regions on the dial can be changed. Depending on the design of the dial, some may allow for the regions to be added or removed, while others may have a fixed number of regions.

## 4. Is there a specific way to spin the dial for more accurate results?

To ensure more accurate results, it is recommended to spin the dial with a consistent force and speed. This will minimize any potential bias in the results.

## 5. Are there any limitations to using a dial with a pointer to n regions?

One limitation of using a dial with a pointer to n regions is that it relies on chance and does not take into account any other factors or variables. Also, the accuracy of the results may be affected if the dial is not well-designed or if the spinning method is inconsistent.

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