import random

def calc_loops(n):
    cnt_loops = 0

    while n > 0:
        ends = []
        for i in range(0, n):
            ends.append(i)
            ends.append(i)
        i = -1
        j = -1
        while i == j:
            i = random.randint(0, len(ends) - 1)
            j = random.randint(0, len(ends) - 1)
        if ends[i] == ends[j]:
            cnt_loops += 1
        n -= 1

    return cnt_loops

# Simulation
n = 100
n_trials = int(5e4)
sum = 0.0
for i in range(0, n_trials):
    sum += calc_loops(n)
print(sum / n_trials)

# Exact computation
res = 0.0
n = 100
while n > 0:
    res += 1.0 / (2 * n - 1)
    n -= 1
print(res)