Grid Probability

Question

There is a 3x3 grid composed of 9 unit squares. Each square can either be Yellow or Blue with equal independent probability. What is the probability that there is at least one 2x2 square containing only Blue squares?

Answer 1

Cases with at least one 2x2 square:
There are $2^5$ outcomes where the lower right 2x2 is blue.
There are $2^5$ outcomes where the upper right 2x2 is blue.
There are $2^5$ outcomes where the lower left 2x2 is blue.
There are $2^5$ outcomes where the upper left 2x2 is blue.

Cases with at least two 2x2 squares:
There are four pairs of 2x2 squares which leave 3 squares untouched. That is $4\cdot2^3$ cases.
There are two pairs of 2x2 squares which leave 2 squares untouched. That is $2\cdot2^2$ cases.

Cases with at least three 2x2 square:
There are four triples of 2x2 squares which leave 1 squares untouched. That is $4\cdot2^1$ cases.

Cases with at least four 2x2 square:
There is 1 case.

Inclusion/Exclusion gives the number of succesful outcomes:
$(4\cdot 2^5)-(4\cdot 2^3+2\cdot 2^2)+(4\cdot 2^1)-(1)$

Whereas the total number of outcomes is $2^9$.

Probability: $95/512$