def fac(n): return 1 if n==0 else n * fac(n-1)
def choose(n,k): return fac(n) / fac(k) / fac(n-k) 
def prob(n,k,p): return choose(n,k) * p ** k * (1-p) ** (n-k)

def prob_lt(n,k,p): return sum([prob(n,i,p) for i in range(p)])

SIGS = 30
ACTUALSIGS = 10
POWRETURN = 0.03
POSRETURN = 0.01

# Expected number of signatures on a block
def ev(pos):
    return SIGS * pos

# Chance you have at least k sigs
def at_least_k(pos, k):
    return sum([prob(SIGS, i, pos) for i in range(k, SIGS + 1)])

# Expected number of signatures on a block filtering all <k
def ev_atleast_k(pos, k):
    total, subprob = 0, 0
    for i in range(k, SIGS + 1):
        p = prob(SIGS, i, pos)
        subprob += p
        total += i * p
    return total / subprob

def normal_mining_return(pow, pos):
    return pow * POWRETURN + ev(pos) * POSRETURN / ACTUALSIGS

def attack_mining_return(pow, pos, k):
    powtotal, postotal, subprob = 0, 0, 0
    # Case 1: mined PoW block, PoS at least k instances
    case_1_prob = pow * at_least_k(pos, k)
    subprob += case_1_prob
    postotal += case_1_prob * ev_atleast_k(pos, k) * POSRETURN / ACTUALSIGS
    powtotal += case_1_prob * POWRETURN
    # Case 2: mined PoW block, PoS less than k: discard
    # Case 3: others mined PoW block
    subprob += (1 - pow)
    postotal += (1 - pow) * ev(pos) * POSRETURN / ACTUALSIGS
    return powtotal / subprob + postotal / subprob