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June 20th, 2010, 12:25 AM   #1
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Drawing black tiles! Tough one

Hello everyone.

-Consider that there is a chess like board with 8 * 8 tiles. All the tiles are white!!!

-There is a magic dice which rolls 95% of the time white and 5% of the time black.

-I use this dice only once for every of the 8*8=64 tiles.

-After passing from all the tiles there might be a chance that some tiles changed their color from white to black.

-I would like to find what is the chance after having used this dice after each one to be at least one black square created!

-This is the hard part. I Would like to see one black square somewhere. So this is not I think and i.i.d problem. A black square might be created in A1,A2,B1,B2 or in A2,A3,B2,B3 and so on. I do not know how to try to think for these type of problem.

I would like to have this calculated fro my thesis but as I am not a mathematician student there is almost no support from my department.

I would like to thank you in advance for your help.
Best Regards
Alex.
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June 20th, 2010, 02:25 AM   #2
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Re: Drawing black tiles! Tough one

Hello Alex,

For the square, at least 4 tiles must be turned black. How many options do you have to find such a square? This helps, but Im not sure about the answer.

Hoempa
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June 20th, 2010, 03:00 PM   #3
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Re: Drawing black tiles! Tough one

Exactly this is the hard part of the problem.. What is the chance of having one square created? I do not know this answer.
I must find a way (that is the reason of making the post here) to calculate it.

Best Regards
Alex
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June 25th, 2010, 04:19 PM   #4
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Re: Drawing black tiles! Tough one

By "black square" do you mean a 2x2 block of black tiles? If so, the chance (given that one tile in twenty is black) is about 0.03 %.
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June 30th, 2010, 01:53 AM   #5
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Re: Drawing black tiles! Tough one

Thanks for your answer.
Yes a square is defined as a 2*2 placement. The problem is that 4 squares aligned in a straight line do not count as a square. Also I would like to calculate the chance at least one such box to be created.

If we finally manage to solve this ... then I ll provide an extension to this

Best Regards
Alex.
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July 1st, 2010, 03:17 AM   #6
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Re: Drawing black tiles! Tough one

I used a recurrence and found that of the 2^64 possible patterns, exactly 16477833186525760257 contain at least one 2x2 block. That's about 89.3265 % of them. But if a cell is coloured in black only 5 % of the time, the chance of a 2x2 block somewhere on the board falls to 0.03 %.
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July 2nd, 2010, 07:02 AM   #7
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Re: Drawing black tiles! Tough one

The change that any given 4 square block is coloured black is 0.05^4

There are 49 such blocks on an 8 x 8 board.

Therefore the chance at least 1 is black is black = (1 - 0.05^4)^49 = 0.000306 or 0.0306%
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July 2nd, 2010, 07:17 AM   #8
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Re: Drawing black tiles! Tough one

Quote:
Originally Posted by CarlPierce
The change that any given 4 square block is coloured black is 0.05^4

There are 49 such blocks on an 8 x 8 board.

Therefore the chance at least 1 is black is black = (1 - 0.05^4)^49 = 0.000306 or 0.0306%
That's a reasonable first approximation, but the probabilities aren't independent.
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