MathDB

suppose that

0\le p \le 1

and we have a wooden square with side length

1

. in the first step we cut this square into

4

smaller squares with side length

\frac{1}{2}

and leave each square with probability

p

or take it with probability

1-p

. in the next step we cut every remaining square from the previous step to

4

smaller squares (as above) and take them with probability

1-p

. it's obvios that at the end what remains is a subset of the first square. a) show that there exists a number

0<p_0<1

such that for

p>p_0

the probability that the remainig set is not empty is positive and for

p<p_0

this probability is zero. b) show that for every

p\neq 1

with probability

1

, the remainig set has size zero. c) for this statement that the right side of the square is connected to the left side of the square with a path, write anything that you can.

Problem Statement