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#1 2011-07-21 19:45:57

anandbrar
Member
Registered: 2011-07-20
Posts: 70

Drunken man with a gun...

Hi all,

I just thought up this problem involving probability..

"A drunken man with a gun in his hand has to shoot a thief. He has 7 shots with him; the thief dies if he gets hit with 2 shots. Presently the probability of the drunk scoring a hit is 1/8, but with every shot he gets more sober and the probability increases by 1/8.

Also at 7/8 probability, he has a Super Shot in which the thief will die in a single shot.

What is most likely to happen? - Death with 2 hits, A Super Shot or the thief survives???"

this is my guess..

what do you all think?:cool:


"widen your gaze... extend beyond the obvious..."

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#2 2011-07-21 22:29:59

gAr
Member
Registered: 2011-01-09
Posts: 3,478

Re: Drunken man with a gun...

Hi anandbrar,


Last edited by gAr (2011-07-21 23:03:43)


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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#3 2011-07-21 23:03:14

anandbrar
Member
Registered: 2011-07-20
Posts: 70

Re: Drunken man with a gun...

Hi gAr,


"widen your gaze... extend beyond the obvious..."

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#4 2011-07-21 23:06:50

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 85,215

Re: Drunken man with a gun...

Hi gAr;

My software went kablooey so I could not compute the answers. What did your transition matrix look like?


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#5 2011-07-21 23:18:08

gAr
Member
Registered: 2011-01-09
Posts: 3,478

Re: Drunken man with a gun...

Hi anandbrar,

I saw your answer, but couldn't verify it. How did you solve?
The matrices are cumbersome to do without a CAS. I hope your method to be simpler!

Hi bobbym,

This is my Q matrix:

and R:

The first 7 rows are for single shot.


What happened to your software?

Last edited by gAr (2011-07-21 23:20:03)


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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#6 2011-07-21 23:24:42

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 85,215

Re: Drunken man with a gun...

Hi gAr;

Thanks for providing that.

I do not know what is wrong with it. I wrote the routine myself so it must have a bug in it that did not show up till now.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#7 2011-07-21 23:28:48

gAr
Member
Registered: 2011-01-09
Posts: 3,478

Re: Drunken man with a gun...

Hi bobbym,

Oh, okay.
Hope you find that bug!


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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#8 2011-07-21 23:56:44

anandbrar
Member
Registered: 2011-07-20
Posts: 70

Re: Drunken man with a gun...

Hi gAr,

For the answer in the first post..i just figured that to reach 50% probability, there is a more than 50% probability..like

1/8 is probability to hit, so 7/8 is probability to miss and also to increase chance to 2/8
so 7/8 * 6/8 * 5/8 would be probability to get 4/8.

I solved the following few equations for consecutive hits

1. two consecutive shots 1st and 2nd ->


2. two consecutive shots 2nd and 3rd ->

3.

4. the probability reaches a max here out of the six cases that i tested.

5.

6. the probability has already started falling...needless to put in the sixth case
7. Super Shot :

probability at Super Shot: 7/8 ----- (1)
to reach super shot probability
7/8 * 6/8 * 5/8 * 4/8 * 3/8 * 2/8 ----- (2)

Hence total probability = (1) * (2) = 0.0168

I would be grateful if any loopholes be pointed out..i think there are some..

Thanks in advance!


"widen your gaze... extend beyond the obvious..."

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#9 2011-07-22 00:55:49

gAr
Member
Registered: 2011-01-09
Posts: 3,478

Re: Drunken man with a gun...

Hi anandbrar,

I could verify the propability of death at supershot. I too did the same way.

But your answer seems to be answering a different question.
Is there something else you need?


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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#10 2011-07-22 01:04:55

anandbrar
Member
Registered: 2011-07-20
Posts: 70

Re: Drunken man with a gun...

Hi gAr,
Oh yes I think I was unknowingly answering the question "Death by which shots is most probable?"

Was I right on this one?

Thanks..


"widen your gaze... extend beyond the obvious..."

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#11 2011-07-22 02:07:43

gAr
Member
Registered: 2011-01-09
Posts: 3,478

Re: Drunken man with a gun...

Hi anandbrar,

I guess that question is not clear.
There are many combinations:
Hit, Miss,Miss, Hit
Miss, Miss, Hit, Hit etc.

Do you want to count only the cases where there are 2 consecutive hits?


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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#12 2011-07-22 02:29:30

anandbrar
Member
Registered: 2011-07-20
Posts: 70

Re: Drunken man with a gun...

Hi gAr,

Yes, i just want to count only the cases where there are 2 consecutive hits.


"widen your gaze... extend beyond the obvious..."

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#13 2011-07-22 02:48:19

gAr
Member
Registered: 2011-01-09
Posts: 3,478

Re: Drunken man with a gun...

Hi anandbrar,

Okay, I came to know from your solution.
I was about to update.

Anyway, here's the probabilities I get:


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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#14 2011-07-22 17:31:14

anandbrar
Member
Registered: 2011-07-20
Posts: 70

Re: Drunken man with a gun...

Hi gAr,

Sorry but i didn't get the 4th case?..


"widen your gaze... extend beyond the obvious..."

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#15 2011-07-22 17:52:53

gAr
Member
Registered: 2011-01-09
Posts: 3,478

Re: Drunken man with a gun...

Hi,

It's (Miss)(Miss)(Miss)(Hit)(Hit)


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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#16 2011-07-22 18:07:55

anandbrar
Member
Registered: 2011-07-20
Posts: 70

Re: Drunken man with a gun...

Hi gAr,

No..No.. wait

Oh now i get it...When i first thought of the problem, i had in mind that the probability only increases with a miss and not with a hit..

So, if probability only increases with a miss then the maximum one would be
(miss)(miss)(hit)(hit)


The problem's solved!!!

Thanks gAr, bobbym..


"widen your gaze... extend beyond the obvious..."

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#17 2011-07-22 18:13:20

gAr
Member
Registered: 2011-01-09
Posts: 3,478

Re: Drunken man with a gun...

Hi anandbrar,

If probability only increases with a miss, I think the final answer calculated in previous posts would be different...
I need to check that.


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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#18 2011-07-22 18:30:30

gAr
Member
Registered: 2011-01-09
Posts: 3,478

Re: Drunken man with a gun...

These are the answers I think to be:


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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