prd.py

import random
TEST_N = 10000000


def calc(target):
    rate_max = 1
    rate_min = 0

    while True:
        rate = dt = (rate_max + rate_min)/2.0
        hit = 0
        for i in xrange(TEST_N):
            if random.random() < rate:
                hit += 1
                rate = dt
            rate += dt
        now = hit * 1.0 / TEST_N
        if abs(now - target) < 0.0001:
            return dt
        if now > target:
            rate_max = dt
        else:
            rate_min = dt
        if rate_max < rate_min:
            return dt

rate = calc(0.1)
print "rate:", rate