
Sorry tg,I think you are wrong. If we consider 959 I.e equal to 1110111111..I.e it has more 1's than zeros. I think the correct answer is 1/4.The logic is512=1000000000(9zeros)
1023=1111111111(10 ones).
Exclude the first one we will consider it afterwards.
So there are total 512 terms. Consider 513 and 1022.
513=000000001(8zeros)
1022=111111110(8 ones)
This is exactly complementary.
Note: we have excluded the first one. Same for 514 and 1021,515 and 1020,keep on doing this and you will find 256 numbers have more 1's and 256 has more 0's.now consider the first one. If we add it to both the 256 numbers the more ones will have no change while the ones having more zeros will be half. Half of it will conta have more zeros and half will conta have more ones. So the total numbers having more zeros will be 128 out of 512. So probability is 1/4. 