Advanced Math Question?

Question

Suppose you have a 10 by 10 grid of marbles. There is a 50% chance for each marble that the color is RED and a 50% change the color is BLUE.

What is the probability, given a random configuration, that there will be a 5x5 grid of RED marbles. The 5x5 REDs can be anywhere on the grid.

Answer 1

Ouch! That seems quite difficult. I would say it is quite small. Here is a first approximation:

There are 36 5x5 squares in a 10x10 square. A given 5x5 square has a 2^(-25) chance of being all red. If at most one of these could happen at a time, the the probability would be 36/2^25 which is about .000001. This however is just an upper bound for the probability, since if two 5x5 squares intersect then the probabilities of one being all red is dependent on the coloring of the other.

I can say, however, by linearity of expectation that the expected number of red 5x5's in a randomly colored 10x10 is 36/2^25, so if you were to consider around a billion random 10x10's, you would expect to find around 1000 red 5x5's.

I will be very impressed if someone comes up with the actual probability -- my guess is it would take a computer simulation or a monsterous inclusion/exclusion calculation!

Answer 2

There are 25 different positions for the grid. The top row of the 5x5 is either at rows 1,2,3,4,or 5 while the left most column is either at column 1,2,3,4 or 5. So 5x5=25 different positions for for the grid. The chance that all 25 of those squares come up all RED is (1/2)^25. So the answer is 25*(1/2)^25 or about 7.5*10^(-7)

Answer 3

The 5x5 grid can be in 36 places. The first row of red can be in rows 1,2,3,4,5, and 6....same with the first column of red. So the answer is 36* (.5^25) = .000001073

Answer 4

Only 4 grids can be 5x5
Just a guess: 4/(10^10)