Are you interested in the experiment design or the distribution of middle bits? 1. For two random k-bit numbers, the middle bit distribution gets closer to 50-50 as k gets larger, I believe. 2. If you are asking about the distribution of middle bits for a fixed number M and its product with a randomly chosen set of other numbers (the set of all member ids), that will depend on the distribution of bits in the member ids. If there aren't many of those ids, the imbalance could be significant. 3. If you want to make sure each experiment has a different set of members assigned to each group, you'll need to choose randomly from all such sets. N and the number of bits in N seem irrelevant. Hilarie Quoting Marc LeBrun <mlb@well.com>:

Can we show the middle bit of a product of a random integer with another integer is random?

Motivating application: We're doing a long series of experiments where in each experiment we need to randomly assign statistically half the members of a population to a control group and the other half to a test group. We don't want the assignments to be correlated across different experiments.

Further, we happen to have a random K-bit integer N assigned to each member.

How many independent experiments can these N help us do?

Obviously we can do K, by extracting the M-th bit of N, 0<=M<=K, to define group assignments in the M-th experiment.

And of course we can do 2^K-1, using the parity of the bitwise AND of M & N, 1<=M<=2^K-1. XORing a bunch of random bits seems clearly unbiased.

Now, as an alternative to the AND-and-parity calculation, can we instead justify just taking the "middle bit" of the double-wide product M * N?

The middle bit is essentially the XOR of N and bit-reversed M, plus carries.

So the XOR contribution is the same as with the AND-and-parity approach, but can carries create correlations?