Is this cool with you?

gAr · 2011-04-08 19:47:42

Yes, that's the reason I gave a possible solution of combining A and B in the states in #467.
Start with A6B6, or let it be 66 for short, it goes to 65 or to DB. From 65 to 55 or DA. From 55 to 55,54,45,DA or DB.
So, from 55 through 12, each state will have 5 possible transitions. Finally, 11 will have 4 possible transitions including BA.

*edit: Whoops, some more states: 50,40,30,20,10,05,04,03,02,01

Last edited by gAr (2011-04-08 20:04:51)

bobbym · 2011-04-08 20:03:19

Hi gAr;

What makes it go from 66 to 65? It seems that would be the loss of a bullet. How can we change states if no bullet is fired? The states are labeled according to how many bullets A or B has. 6 means 6 bullets. If you change states on empty chambers then it is possible to end the game with no bullets being fired. Especially with A2, A1, B2, B1 who most likely will not kill anyone. Should not the game end on a death or no more bullets?

Supposing we say that when A pulls the trigger and it is empty that he must pass it to the next B. Then the longest number of trigger pulls is 12. Less when someone is killed but never more than 12.

gAr · 2011-04-08 20:09:50

66 to 65 is same as A6 to B5 - the shot misses the target.

From 55 to 55,54,45,DA or DB

It remains in that state if no shot is fired.

bobbym · 2011-04-08 20:19:16

Have you made a transition matrix out of that?

gAr · 2011-04-08 20:25:06

I'm preparing one.
May take time, 41 states!

I'll take a 40 mins break, see you later.

bobbym · 2011-04-08 20:34:40

Okay gAr, have a good break! I am going to relax a little too.

gAr · 2011-04-08 23:54:35

Hi bobbym,

The process for inverse calculation is running for more than 20 minutes, I don't know when it'll finish.
Here is the P matrix.

#

#
It is not displaying the latex code above, here is the plain text of the matrix:

[0,1/2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1/2,0],
[0,0,1/2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1/2],
[0,0,1/36,5/72,0,0,0,0,5/72,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5/12,5/12],
[0,0,0,1/18,1/18,0,0,0,0,5/36,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5/12,1/3],
[0,0,0,0,1/12,1/24,0,0,0,0,5/24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5/12,1/4],
[0,0,0,0,0,1/9,1/36,0,0,0,0,5/18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5/12,1/6],
[0,0,0,0,0,0,5/36,1/72,0,0,0,0,25/72,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5/12,1/12],
[0,0,0,0,0,0,0,1/6,0,0,0,0,0,5/12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5/12,0],
[0,0,0,0,0,0,0,0,1/18,5/36,0,0,0,0,1/18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1/3,5/12],
[0,0,0,0,0,0,0,0,0,1/9,1/9,0,0,0,0,1/9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1/3,1/3],
[0,0,0,0,0,0,0,0,0,0,1/6,1/12,0,0,0,0,1/6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1/3,1/4],
[0,0,0,0,0,0,0,0,0,0,0,2/9,1/18,0,0,0,0,2/9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1/3,1/6],
[0,0,0,0,0,0,0,0,0,0,0,0,5/18,1/36,0,0,0,0,5/18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1/3,1/12],
[0,0,0,0,0,0,0,0,0,0,0,0,0,1/3,0,0,0,0,0,1/3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1/3,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1/12,5/24,0,0,0,0,1/24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1/4,5/12],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1/6,1/6,0,0,0,0,1/12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1/4,1/3],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1/4,1/8,0,0,0,0,1/8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1/4,1/4],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1/3,1/12,0,0,0,0,1/6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1/4,1/6],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5/12,1/24,0,0,0,0,5/24,0,0,0,0,0,0,0,0,0,0,0,0,0,1/4,1/12],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1/2,0,0,0,0,0,1/4,0,0,0,0,0,0,0,0,0,0,0,0,1/4,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1/9,5/18,0,0,0,0,1/36,0,0,0,0,0,0,0,0,0,0,0,1/6,5/12],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2/9,2/9,0,0,0,0,1/18,0,0,0,0,0,0,0,0,0,0,1/6,1/3],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1/3,1/6,0,0,0,0,1/12,0,0,0,0,0,0,0,0,0,1/6,1/4],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4/9,1/9,0,0,0,0,1/9,0,0,0,0,0,0,0,0,1/6,1/6],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5/9,1/18,0,0,0,0,5/36,0,0,0,0,0,0,0,1/6,1/12],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2/3,0,0,0,0,0,1/6,0,0,0,0,0,0,1/6,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5/36,25/72,0,0,0,0,1/72,0,0,0,0,0,1/12,5/12],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5/18,5/18,0,0,0,0,1/36,0,0,0,0,1/12,1/3],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5/12,5/24,0,0,0,0,1/24,0,0,0,1/12,1/4],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5/9,5/36,0,0,0,0,1/18,0,0,1/12,1/6],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25/36,5/72,0,0,0,0,5/72,0,1/12,1/12],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5/6,0,0,0,0,0,1/12,1/12,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1/6,5/12,0,0,0,0,0,5/12],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1/3,1/3,0,0,0,0,1/3],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1/2,1/4,0,0,0,1/4],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2/3,1/6,0,0,1/6],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5/6,1/12,0,1/12],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1]

You may be knowing of more efficient algorithms.

Last edited by gAr (2011-04-08 23:58:16)

bobbym · 2011-04-09 02:28:07

Hi gAr;

I was able to latex your matrix. But not in one piece. I think there is a limit on the size this latex will support.

Append these 3 rows to the bottom of P.

If you want you can paste this into your post, then I will delete this post here.

gAr · 2011-04-09 02:46:40

Hi bobbym,

It's okay, it looks huge.
Maybe I'll post only the final probabilities I get.
It's slightly different from what we got.

Both alive: 3.17065746753247e-6
A lives: 0.624998414671266
B lives: 0.374998414671266

gAr · 2011-04-09 03:06:50

Maybe it's wrong, getting low values for expected number of shots.

bobbym · 2011-04-09 03:09:01

Hi gAr;

Are you happy with both alive being so low?

gAr · 2011-04-09 03:44:35

Hi bobbym,

Probabilities are ok, but expected number of shots not ok.
Or were you sarcastic by not reading "e-6" at the end?!

bobbym · 2011-04-09 03:56:57

Hi gAr;

Not at all. I was just asking that probability is almost zero. I would thought it would be higher. The states A3,A2,A1 and B3,B2,B1 miss alot.

gAr · 2011-04-09 04:10:47

Hi bobbym,

Yes, it should have been higher.
I'm out of ideas!

bobbym · 2011-04-09 08:54:52

Hi gAr;

It is a hard problem.

Do you have an estimate as to what you think the expected numbers should be?

gAr · 2011-04-09 15:05:59

Hi bobbym,

Ok.
I expect the number of trigger pulls to be less than 30.

bobbym · 2011-04-09 16:04:31

Hi gAr;

I expect it to be a around 3 or maybe 4. The A6 A5 B6 B5 are just going to be too efficient in hitting the target.

gAr · 2011-04-09 16:24:53

Hi bobbym,

Ah, yes!
I forgot it is the average. I gave an estimate of highest number such that no one is shot.

gAr · 2011-04-09 17:03:12

When only one person gets the chance to fire, I find the expected number to be 5233/2310.

And hmmm,
I'm beginning to think that the matrix in #509 may be right.
When both can shoot, the expected number should get less, isn't it?

The time to absorption is:

Last edited by gAr (2011-04-09 18:05:31)

bobbym · 2011-04-09 18:23:40

Hi gAr;

I agree, it should be less for two of them shooting.

gAr · 2011-04-09 18:41:15

Hi bobbym,

The first markov chain gives 2.1710782777134774, this is also agreeable.
Is a simulation possible for this problem?

bobbym · 2011-04-09 21:26:08

Yes: But I do not understand your model well enough to do it.

I am going to get a little rest. If you want I will do it first thing when I get up. Can you give me in words what happens on each possibility? I need to know exactly what happens in your model on a hit, a miss and and a empty chamber. Try to simplify it because simple models run faster and are easier to program.

I think that we will get different answers depending on how we define the three possibilities ( hit, miss, empty chamber ). So lets start with yours and see if we can get the math to agree.

I have built another model that has sligtly different rules than your big one and I am getting 2.14555 as the expected number of pulls.

gAr · 2011-04-09 22:09:53

Hi,

Let's forget that big matrix, it's probably wrong.
I'm trying to approach it in other ways.
Your answer is very close to the post above that.

See you later.

bobbym · 2011-04-10 04:19:59

Hi gAr;

I am beginning to think that how we handle a blank chamber is arbitrary. You can think of it one way or another each will produce different answers.

It has to be defined before answers will agree. The rule for that should also be stated in the problem. Since you posed the problem you have the honor of explicitly defining what should be done on a blank chamber.

gAr · 2011-04-10 05:03:06

Hi bobbym,

Ok.
I'll try to define the problem again clearly:

There are two players with a revolver each, with a capacity of 6 bullets. They start with their revolvers fully loaded.
Each player tries to shoot the other person in turns, A gets the first turn.
They spin the cylinder before shooting. After they pull the trigger, whether the bullet goes off or not, the other player gets the turn if he's not dead yet.
Each of them have a 50 percent probability of hitting the target.

What is the probability that i) A will be dead ii) B will be dead iii) Both are alive ?

Math Is Fun Forum

#501 2011-04-08 19:47:42

Re: Is this cool with you?

#502 2011-04-08 20:03:19

Re: Is this cool with you?

#503 2011-04-08 20:09:50

Re: Is this cool with you?

#504 2011-04-08 20:19:16

Re: Is this cool with you?

#505 2011-04-08 20:25:06

Re: Is this cool with you?

#506 2011-04-08 20:34:40

Re: Is this cool with you?

#507 2011-04-08 23:54:35

Re: Is this cool with you?

#508 2011-04-09 02:28:07

Re: Is this cool with you?

#509 2011-04-09 02:46:40

Re: Is this cool with you?

#510 2011-04-09 03:06:50

Re: Is this cool with you?

#511 2011-04-09 03:09:01

Re: Is this cool with you?

#512 2011-04-09 03:44:35

Re: Is this cool with you?

#513 2011-04-09 03:56:57

Re: Is this cool with you?

#514 2011-04-09 04:10:47

Re: Is this cool with you?

#515 2011-04-09 08:54:52

Re: Is this cool with you?

#516 2011-04-09 15:05:59

Re: Is this cool with you?

#517 2011-04-09 16:04:31

Re: Is this cool with you?

#518 2011-04-09 16:24:53

Re: Is this cool with you?

#519 2011-04-09 17:03:12

Re: Is this cool with you?

#520 2011-04-09 18:23:40

Re: Is this cool with you?

#521 2011-04-09 18:41:15

Re: Is this cool with you?

#522 2011-04-09 21:26:08

Re: Is this cool with you?

#523 2011-04-09 22:09:53

Re: Is this cool with you?

#524 2011-04-10 04:19:59

Re: Is this cool with you?

#525 2011-04-10 05:03:06

Re: Is this cool with you?

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