##
## Lets numerically look at what direction we should try to prove: 
##
ns = seq(15, 20)
P_n_tails_in_n = 0.5^ns

P_less_than_n_heads_in_4n = pbinom(ns-1, 4*ns, 1/2)

first_gt_second = P_n_tails_in_n > P_less_than_n_heads_in_4n

df = data.frame(n=ns, P_n_tails_in_n=P_n_tails_in_n, P_less_than_n_heads_in_4n=P_less_than_n_heads_in_4n, first_gt_second=first_gt_second)
print(df)