4. Take a random 8-bit number consisting of 0s and 1s, the probability that the sum of the digits in the even positions is equal to the sum of the digits in the odd positions is
$\qquad$ .
$S_{\text {odd }}=1$ has 4 permutations; $S_{\text {odd }}=2$ has $C_{4}^{2} \cdot C_{3}^{1}=18$ permutations; $S_{\text {odd }}=3$ has $C_{4}^{3} \cdot C_{3}^{2}=12$ permutations; $S_{\text {odd }}=4$ has 1 permutation.
Let $A=$ “the sum of the digits in even positions is equal to the sum of the digits in odd positions”, then $P(A)=\frac{35}{128}$.