import math

def binomial_coefficient(n, k):
    return math.factorial(n) // (math.factorial(k) * math.factorial(n - k))

def probability_single_person(n, k, p):
    probability = 0
    for i in range(k, n + 1):
        probability += binomial_coefficient(n, i) * (p ** i) * ((1 - p) ** (n - i))
    return probability

def probability_at_least_one_person(num_people, flips_per_person, min_heads, p):
    single_prob = probability_single_person(flips_per_person, min_heads, p)
    return 1 - (1 - single_prob) ** num_people

# Parameters for different strategies
num_people = 100 # number of people
flips_per_person = 10 # number of coin flips per person
min_heads = 9 # minimum number of heads for success

# Probability for a fair coin (pure chance)
p_fair = 0.5
probability_fair = probability_at_least_one_person(num_people, flips_per_person, min_heads, p_fair)
print(f"Probability with fair coin: {probability_fair:.4f} or {probability_fair * 100:.2f}%")

# Probability for a skilled trader with better odds (e.g., 60% success rate)
p_skilled = 0.6
probability_skilled = probability_at_least_one_person(num_people, flips_per_person, min_heads, p_skilled)
print(f"Probability with skilled trader: {probability_skilled:.4f} or {probability_skilled * 100:.2f}%")

# Probability for a hedged strategy (e.g., 70% success rate)
p_hedged = 0.7
probability_hedged = probability_at_least_one_person(num_people, flips_per_person, min_heads, p_hedged)
print(f"Probability with hedged strategy: {probability_hedged:.4f} or {probability_hedged * 100:.2f}%")

# Probability for an ETF with diversified risk (e.g., 55% success rate)
p_etf = 0.55
probability_etf = probability_at_least_one_person(num_people, flips_per_person, min_heads, p_etf)
print(f"Probability with ETF: {probability_etf:.4f} or {probability_etf * 100:.2f}%")