def normal_probability_above(lo, mu=0, sigma=1):
    return 1 - normal_cdf(lo, mu, sigma)
def normal_probability_between(lo, hi, mu=0, sigma=1):
    return normal_cdf(hi, mu, sigma) - normal_cdf(lo, mu, sigma)
def normal_probability_below(lo, mu=0, sigma=1):
    return normal_cdf(lo, mu, sigma)