Ah, I misunderstood the problem as first. I'll think about this. In the mean time, I ran some simulations (10000 trials of n flips)
n=3, p=.2512 vs .25 actual
n=4, p=.1867 vs 0.1875 actual
n=5, p=.1881
n=6, p=.1604
n=7, p=.1561
...
n=10, p=.1249
...
n=20, p=.0915
...
n=100, p=.0403