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**mathstudent2000****Member**- Registered: 2013-07-26
- Posts: 79

1. A point is always at 0, 1, or 2. At each step, the point moves 1 unit, either up or down. If the point is at 1, it moves to either 0 or 2 with equal probability. However, if the point is at 2, it must move to 1 with its next step.

For a point currently at 1, let t_1 be the expected number of steps the point will take before it reaches 0 for the first time. Similarly, let t_2 be the expected number of steps for the point to first reach 0 if the point instead starts at 2.

(a) Consider the next step for a point that is currently at 2. Express t_2 in terms of t_1.

(b) Now consider the next step for a point that is currently at 1. Give an expression for t_1 in terms of t_2 that is not equivalent to (a).

(c) Use your answers to (a) and (b) to find t_1 and t_2.

2. Prove that (2n combination n) is always even for n>0.

In this problem we will discover and prove another identity.

(a) Evaluate:

(i) {1\choose 0}^2 + {1\choose 1}^2.

(ii) {2\choose 0}^2 + {2\choose 1}^2 + {2\choose 2}^2

(iii) {3\choose 0}^2 + {3\choose 1}^2 + {3\choose 2}^2 + {3\choose 3}^2.

(iv) {4\choose 0}^2 + {4\choose 1}^2 + {4\choose 2}^2 + {4\choose 3}^2 + {4\choose 4}^2.

(b) Do your answers in part (a) appear on Pascal's Triangle? If so, where?

(c) Guess an identity based on your observations in parts (a) and (b).

(d) Test your identity by putting in n=1, 2, 3, 4 and making sure you get the relationships you found in part (a).

(e) Prove your relationship using a block walking argument.

(f) Prove your relationship using a committee-forming argument.

Genius is one percent inspiration and ninety-nine percent perspiration

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 94,912

Hi;

Yikes, that is half a semester of work!

You use an absorbing Markov chain with that. Did you set one up for the problem? Do you know how to do that?

**In mathematics, you don't understand things. You just get used to them.**

**If it ain't broke, fix it until it is.**

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**mathstudent2000****Member**- Registered: 2013-07-26
- Posts: 79

I don't really know what is an absorbing Markov chain. Do you have a way to solve it using only algebra and combinations?

Genius is one percent inspiration and ninety-nine percent perspiration

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 94,912

Hi;

What grade are you in? Is this an assignment and what textbook are you working out of?

**In mathematics, you don't understand things. You just get used to them.**

**If it ain't broke, fix it until it is.**

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**mathstudent2000****Member**- Registered: 2013-07-26
- Posts: 79

I'm in 8th grade and this is a question for my AOPS Counting and Probability Hw.

Genius is one percent inspiration and ninety-nine percent perspiration

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 94,912

Hi;

I think that 1 is an absorbing Markov chain and I do not have any other way of solving it.

What method do they suggest over there?

I am glad to help but these are questions for an online homework assignment. You really should try them on your own.

When you post some of your own work then I can help with the rest.

**In mathematics, you don't understand things. You just get used to them.**

**If it ain't broke, fix it until it is.**

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**mathstudent2000****Member**- Registered: 2013-07-26
- Posts: 79

they suggest using any sort of probability

Genius is one percent inspiration and ninety-nine percent perspiration

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 94,912

The AOPS is somewhat advanced and does require a knowledge of lots of math. Why are you doing it as an 8th grader, if I may ask?

**In mathematics, you don't understand things. You just get used to them.**

**If it ain't broke, fix it until it is.**

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**mathstudent2000****Member**- Registered: 2013-07-26
- Posts: 79

i am doing geometry and algebra 2 and will be starting trig soon

Genius is one percent inspiration and ninety-nine percent perspiration

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 94,912

Those questions are way beyond that. You age is about 13 - 14?

**In mathematics, you don't understand things. You just get used to them.**

**If it ain't broke, fix it until it is.**

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**mathstudent2000****Member**- Registered: 2013-07-26
- Posts: 79

i'm only 12

Genius is one percent inspiration and ninety-nine percent perspiration

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 94,912

Yikes! Congratulations! You are a young man?

**In mathematics, you don't understand things. You just get used to them.**

**If it ain't broke, fix it until it is.**

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**mathstudent2000****Member**- Registered: 2013-07-26
- Posts: 79

yes, but i started math early and i skipped a few grades in math

Genius is one percent inspiration and ninety-nine percent perspiration

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 94,912

Okay, as soon as collect my notes I will answer as many of those as I can.

**In mathematics, you don't understand things. You just get used to them.**

**If it ain't broke, fix it until it is.**

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**mathstudent2000****Member**- Registered: 2013-07-26
- Posts: 79

ok

Genius is one percent inspiration and ninety-nine percent perspiration

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**mathstudent2000****Member**- Registered: 2013-07-26
- Posts: 79

did you get it?

Genius is one percent inspiration and ninety-nine percent perspiration

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 94,912

Hi;

Not yet, It will take a while. I am helping someone else. Also, those are tougher than the other ones. Please give me the entire day.

**In mathematics, you don't understand things. You just get used to them.**

**If it ain't broke, fix it until it is.**

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**mathstudent2000****Member**- Registered: 2013-07-26
- Posts: 79

ok

Genius is one percent inspiration and ninety-nine percent perspiration

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 94,912

Hi;

For a point currently at 1, let t_1 be the expected number of steps the point will take before it reaches 0 for the first time.

I am getting t1 = 3.

Similarly, let t_2 be the expected number of steps for the point to first reach 0 if the point instead starts at 2.

I am getting t2 = 4.

**In mathematics, you don't understand things. You just get used to them.**

**If it ain't broke, fix it until it is.**

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**mathstudent2000****Member**- Registered: 2013-07-26
- Posts: 79

thank you

Genius is one percent inspiration and ninety-nine percent perspiration

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 94,912

2 a i) 2

2 a ii) 6

2 a iii) 20

2 a iv) 70

2b) Yes, in the center of every odd row.

**In mathematics, you don't understand things. You just get used to them.**

**If it ain't broke, fix it until it is.**

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