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

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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**gAr****Member**- Registered: 2011-01-09
- Posts: 3,479

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

Hi gAr,

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

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 92,942

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.**

**I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.**

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**gAr****Member**- Registered: 2011-01-09
- Posts: 3,479

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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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 92,942

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.**

**I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.**

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**gAr****Member**- Registered: 2011-01-09
- Posts: 3,479

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

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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**gAr****Member**- Registered: 2011-01-09
- Posts: 3,479

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?

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

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

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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**gAr****Member**- Registered: 2011-01-09
- Posts: 3,479

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?

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

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

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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**gAr****Member**- Registered: 2011-01-09
- Posts: 3,479

Hi anandbrar,

Okay, I came to know from your solution.

I was about to update.

Anyway, here's the probabilities I get:

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

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

Hi gAr,

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

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

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**gAr****Member**- Registered: 2011-01-09
- Posts: 3,479

Hi,

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

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

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

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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**gAr****Member**- Registered: 2011-01-09
- Posts: 3,479

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.

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

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**gAr****Member**- Registered: 2011-01-09
- Posts: 3,479

These are the answers I think to be:

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

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