Math Is Fun Forum

  Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °

You are not logged in.

#1 2016-03-15 23:46:25

jakndaxta
Member
Registered: 2016-03-15
Posts: 18

[Probability, University Introductory Level] "Rat Run" Question

A rat is released in the space outside a maze consisting of three rooms and six doors,
as depicted in the following figure.

Rat Run Maze

Whenever the rat is in a space or room with k doors, it chooses each of these doors to move through next with probability 1/k. We are interested in the movement of the rat from when it first enters the maze until it first leaves.
(a) If the rat enters the maze at Room 1, find the probability that it will leave
from Room 3.
(b) If the rat starts in the space around the maze, find the probability that it will
eventually leave the maze from Room 3.
(c) If the rat leaves the maze from Room 3 find the probability that it entered at
Room 1.
(d) Suppose that the rat is now in the maze and we gain information which
makes us 70% confident that it entered at Room 1 and 20% confident
that it entered at Room 2, find the probability that:
(d.i) the rat will leave from Room 3
(d.ii) the rat entered at Room 1 if it leaves from Room 3
(d.iii) the rat entered at Room 1 if it leaves from Room 1.
For (d) it may be assumed that had we known at which room the rat entered the maze,
the said additional information would not alter our beliefs regarding subsequent movements of the rat.

Hi guys I'm currently trying to figure this question out, but I have no idea how to approach it. I came across a previous post on this question that utilised Markov Chains to calculate the probability of part (a) but I don't understand how its done.

I'd appreciate it if someone could walk me through completing at least part (a) and hopefully the rest of the question as well.

Thank you for your time!

Offline

#2 2016-03-16 18:36:08

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: [Probability, University Introductory Level] "Rat Run" Question

Hi;

Have you studied Markov chains yet?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline

#3 2016-03-16 22:18:55

jakndaxta
Member
Registered: 2016-03-15
Posts: 18

Re: [Probability, University Introductory Level] "Rat Run" Question

bobbym wrote:

Hi;

Have you studied Markov chains yet?

Hi bobbym,

Unfortunately I have not, I looked through your example the last time you did this problem and I was having trouble seeing how it could apply to this problem, would it be okay if you guided me through step by step? I'd really like an intuitive understanding of how to do this question.

Offline

#4 2016-03-17 02:56:22

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: [Probability, University Introductory Level] "Rat Run" Question

I am basically a problem solver, (though not a good one), a teacher I am not. Are you sure you would not rather learn this technique using a somewhat simpler example?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline

#5 2016-03-17 03:15:51

jakndaxta
Member
Registered: 2016-03-15
Posts: 18

Re: [Probability, University Introductory Level] "Rat Run" Question

bobbym wrote:

I am basically a problem solver, (though not a good one), a teacher I am not. Are you sure you would not rather learn this technique using a somewhat simpler example?

If you had the time I would love for a simpler example to help increase my understanding before showing how it applies to this problem, but if you do not have time I would rather you showed me the way it applies to this question straight up. Thanks again!

Offline

#6 2016-03-17 06:22:51

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: [Probability, University Introductory Level] "Rat Run" Question

We start with a maze of 4 rooms, each painted a different color and labeled 1,2,3 and 4. We wish to release a mouse into one of the rooms and observe its behavior. We would like to be able to predict the mouses movements in the maze.

LDJdMDY.png


Understand so far?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline

#7 2016-03-17 19:36:05

jakndaxta
Member
Registered: 2016-03-15
Posts: 18

Re: [Probability, University Introductory Level] "Rat Run" Question

bobbym wrote:

We start with a maze of 4 rooms, each painted a different color and labeled 1,2,3 and 4. We wish to release a mouse into one of the rooms and observe its behavior. We would like to be able to predict the mouses movements in the maze.

http://i.imgur.com/LDJdMDY.png


Understand so far?

Looks good so far!

Offline

#8 2016-03-18 03:58:05

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: [Probability, University Introductory Level] "Rat Run" Question

We now have a 4 rooms and a mouse, this is called the initial state.

We want to select a room for the mouse, if we agree that we would want to do this in an equally likely way then we form a row vector [1 / 4, 1 / 4, 1 / 4, 1 / 4]. If we always wanted the mouse to start in room 1  then we would form the row vector [1,0,0,0]. This vector is called the initial state vector.

Okay?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline

#9 2016-03-18 15:38:33

jakndaxta
Member
Registered: 2016-03-15
Posts: 18

Re: [Probability, University Introductory Level] "Rat Run" Question

bobbym wrote:

We now have a 4 rooms and a mouse, this is called the initial state.

We want to select a room for the mouse, if we agree that we would want to do this in an equally likely way then we form a row vector [1 / 4, 1 / 4, 1 / 4, 1 / 4]. If we always wanted the mouse to start in room 1  then we would form the row vector [1,0,0,0]. This vector is called the initial state vector.

Okay?

Sounds good smile

Offline

#10 2016-03-18 16:33:09

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: [Probability, University Introductory Level] "Rat Run" Question

To further clarify, the mouse can only move right - left or left-right and up-down or down up. No diagonal moves such as 1 to 4.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline

#11 2016-03-19 01:07:05

jakndaxta
Member
Registered: 2016-03-15
Posts: 18

Re: [Probability, University Introductory Level] "Rat Run" Question

bobbym wrote:

To further clarify, the mouse can only move right - left or left-right and up-down or down up. No diagonal moves such as 1 to 4.

Does this account for the outside box then? As all boxes can be moved between directly (except for boxes 1 and 3).

Offline

#12 2016-03-19 01:29:38

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: [Probability, University Introductory Level] "Rat Run" Question

The outside box is just a product of the cut and paste.

What is important is how the mouse moves through the rooms. He can go from 1 to 2 or 1 to 3 but not 1 to 4. He can also go from 3 to 4 and 3 to 1 but not 3 to 2 etc.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline

#13 2016-03-19 15:57:56

jakndaxta
Member
Registered: 2016-03-15
Posts: 18

Re: [Probability, University Introductory Level] "Rat Run" Question

bobbym wrote:

The outside box is just a product of the cut and paste.

What is important is how the mouse moves through the rooms. He can go from 1 to 2 or 1 to 3 but not 1 to 4. He can also go from 3 to 4 and 3 to 1 but not 3 to 2 etc.

I meant the outside box in the initial maze diagram, the "fourth" room if you will

Offline

#14 2016-03-19 19:52:23

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: [Probability, University Introductory Level] "Rat Run" Question

The yellow room? I do not understand.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline

#15 2016-03-21 00:20:45

jakndaxta
Member
Registered: 2016-03-15
Posts: 18

Re: [Probability, University Introductory Level] "Rat Run" Question

showimage.php?pid=257669&filename=Screen+Shot+2013-03-18+at+8.10.36+PM.png

I mean the space surrounding boxes 1, 2 and 3.

Offline

#16 2016-03-21 04:13:57

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: [Probability, University Introductory Level] "Rat Run" Question

You are getting ahead of yourself, are you ready with the example I have given to move on?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline

#17 2016-03-21 22:18:29

jakndaxta
Member
Registered: 2016-03-15
Posts: 18

Re: [Probability, University Introductory Level] "Rat Run" Question

bobbym wrote:

You are getting ahead of yourself, are you ready with the example I have given to move on?

No worries, yes I am!

Offline

#18 2016-03-22 06:23:10

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: [Probability, University Introductory Level] "Rat Run" Question

We decide we want to start with the mouse in the red room (room 1) so the initial vector will be

The experiment now consists of observing at regular fixed intervals of time the position and movement of the mouse. For example, after 1 such interval, the mouse could have stayed where he was or moved to either room 2 or room 3.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline

#19 2016-03-24 21:47:08

jakndaxta
Member
Registered: 2016-03-15
Posts: 18

Re: [Probability, University Introductory Level] "Rat Run" Question

bobbym wrote:

We decide we want to start with the mouse in the red room (room 1) so the initial vector will be

The experiment now consists of observing at regular fixed intervals of time the position and movement of the mouse. For example, after 1 such interval, the mouse could have stayed where he was or moved to either room 2 or room 3.

Sure thing smile

Offline

#20 2016-03-25 03:52:58

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: [Probability, University Introductory Level] "Rat Run" Question

Of course when we are speaking of the future we can only do so probabilistically.

We now have to assign probabilities for each room

From Room 1 to  room 1 we assign 1 / 3 to room 2 we assign 1 / 3 and to room 3 we assign 1 / 3.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline

#21 2016-03-25 20:17:46

jakndaxta
Member
Registered: 2016-03-15
Posts: 18

Re: [Probability, University Introductory Level] "Rat Run" Question

bobbym wrote:

Of course when we are speaking of the future we can only do so probabilistically.

We now have to assign probabilities for each room

From Room 1 to  room 1 we assign 1 / 3 to room 2 we assign 1 / 3 and to room 3 we assign 1 / 3.

Sounds good!

Offline

#22 2016-03-25 20:28:04

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: [Probability, University Introductory Level] "Rat Run" Question

From Room 2 to  room 1 we assign 1 / 3 to room 2 we assign 1 / 3 and to room 4 we assign 1 / 3.

From Room 3 to  room 1 we assign 1 / 3 to room 3 we assign 1 / 3 and to room 4 we assign 1 / 3.

From Room 4 to  room 2 we assign 1 / 3 to room 3 we assign 1 / 3 and to room 4 we assign 1 / 3.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline

#23 2016-03-29 12:01:51

jakndaxta
Member
Registered: 2016-03-15
Posts: 18

Re: [Probability, University Introductory Level] "Rat Run" Question

bobbym wrote:

From Room 2 to  room 1 we assign 1 / 3 to room 2 we assign 1 / 3 and to room 4 we assign 1 / 3.

From Room 3 to  room 1 we assign 1 / 3 to room 3 we assign 1 / 3 and to room 4 we assign 1 / 3.

From Room 4 to  room 2 we assign 1 / 3 to room 3 we assign 1 / 3 and to room 4 we assign 1 / 3.

Makes sense so far smile

Offline

#24 2016-03-29 12:03:52

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: [Probability, University Introductory Level] "Rat Run" Question

Are you familiar with matrix operations?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline

#25 2016-03-30 02:03:53

jakndaxta
Member
Registered: 2016-03-15
Posts: 18

Re: [Probability, University Introductory Level] "Rat Run" Question

bobbym wrote:

Are you familiar with matrix operations?

Yes i am smile

Offline

Board footer

Powered by FluxBB