exact

from collections import defaultdict
heads, tails= defaultdict(int, {0: 1}), defaultdict(int, {0: 1})
n=101
for i in range(2, n):
    new_heads, new_tails = defaultdict(int), defaultdict(int)
    for key, value in tails.items():
        new_heads[key] = value
        new_tails[key] = value
    for key, value in heads.items():
        new_heads[key + 1] += value
        new_tails[key - 1] += value
    heads, tails = new_heads, new_tails
    merged = [*heads.items(), *tails.items()]
    awins = sum(value for key, value in merged if key > 0)
    bwins = sum(value for key, value in merged if key < 0)
    ties = sum(value for key, value in merged if key == 0)
    print(f"n: {i}, awin = {awins/2**i*100}, bwin = {bwins/2**i*100}, tie = {ties/2**i*100}")