import math


def factorial(n):
	"""
	Computes the factorial of a number.
	:param n: The number to compute the factorial of.
	:return: Returns the factorial of the number.
	"""
	if n == 0:
		return 1
	for i in range(1, n):
		n *= i
	return n


def combination(n, r):
	"""
	Computes the combination of n choose r.
	:param n: The number of items.
	:param r: The number of items to choose.
	:return: Returns the number of ways to choose r items from n items.
	"""
	return factorial(n) / (factorial(r) * factorial(n - r))


def bnd(x, n, p):
	"""
	Computes the binomial distribution.
	:param x: Number of successes.
	:param n: Number of trials.
	:param p: Probability of success.
	:return: Returns the probability of getting x successes in n trials.
	"""
	return combination(n, x) * p ** x * (1 - p) ** (n - x)


def bnd_mean(n, p):
	"""
	Computes the mean of the binomial distribution.
	:param n: Number of trials.
	:param p: Probability of success.
	:return: Returns the mean of the binomial distribution.
	"""
	return n * p


def bnd_var(n, p):
	"""
	Computes the variance of the binomial distribution.
	:param n: Number of trials.
	:param p: Probability of success.
	:return: Returns the variance of the binomial distribution.
	"""
	return n * p * (1 - p)


def bnd_std(n, p):
	"""
	Computes the standard deviation of the binomial distribution.
	:param n: Number of trials.
	:param p: Probability of success.
	:return: Returns the standard deviation of the binomial distribution.
	"""
	return bnd_var(n, p) ** 0.5


def bnd_leq(x, n, p):
	"""
	Computes the cumulative probability less than or equal to x successes in n trials.
	:param x: Number of successes.
	:param n: Number of trials.
	:param p: Probability of success.
	:return: Returns the cumulative probability less than or equal to x successes in n trials.
	"""
	return sum(bnd(i, n, p) for i in range(x + 1))


def bnd_lt(x, n, p):
	"""
	Computes the cumulative probability less than x successes in n trials.
	:param x: Number of successes.
	:param n: Number of trials.
	:param p: Probability of success.
	:return: Returns the cumulative probability less than x successes in n trials.
	"""
	return sum(bnd(i, n, p) for i in range(x))


def bnd_geq(x, n, p):
	"""
	Computes the cumulative probability greater than or equal to x successes in n trials.
	:param x: Number of successes.
	:param n: Number of trials.
	:param p: Probability of success.
	:return: Returns the cumulative probability greater than or equal to x successes in n trials.
	"""
	return 1 - bnd_lt(x, n, p)


def bnd_gt(x, n, p):
	"""
	Computes the cumulative probability greater than x successes in n trials.
	:param x: Number of successes.
	:param n: Number of trials.
	:param p: Probability of success.
	:return: Returns the cumulative probability greater than x successes in n trials.
	"""
	return 1 - bnd_leq(x, n, p)


def gd(x, p, q=None):
	"""
	Computes the geometric distribution.
	:param x: Number of trials until the first success.
	:param p: Probability of success.
	:param q: Probability of failure.
	:return: Returns the probability of getting the first success on the xth trial.
	"""
	if q is None:
		q = 1 - p
	return q ** (x - 1) * p


def gd_mean(p):
	"""
	Computes the mean of the geometric distribution.
	:param p: Probability of success.
	:return: Returns the mean of the geometric distribution.
	"""
	return 1 / p


def gd_var(p):
	"""
	Computes the variance of the geometric distribution.
	:param p: Probability of success.
	:return: Returns the variance of the geometric distribution.
	"""
	return (1 - p) / p ** 2


def gd_std(p):
	"""
	Computes the standard deviation of the geometric distribution.
	:param p: Probability of success.
	:return: Returns the standard deviation of the geometric distribution.
	"""
	return gd_var(p) ** 0.5


def gd_leq(x, p, q=None):
	"""
	Computes the cumulative probability of getting upto x trials until the first success.
	:param x: Number of trials until the first success.
	:param p: Probability of success.
	:param q: Probability of failure.
	:return: Returns the cumulative probability of getting upto x trials until the first success.
	"""
	if q is not None:
		return sum(gd(i, p, q) for i in range(1, x + 1))
	return sum(gd(i, p) for i in range(1, x + 1))


def gd_lt(x, p, q=None):
	"""
	Computes the cumulative probability of getting less than x trials until the first success.
	:param x: Number of trials until the first success.
	:param p: Probability of success.
	:param q: Probability of failure.
	:return: Returns the cumulative probability of getting less than x trials until the first success.
	"""
	if q is not None:
		return sum(gd(i, p, q) for i in range(1, x))
	return sum(gd(i, p) for i in range(1, x))


def gd_geq(x, p, q=None):
	"""
	Computes the cumulative probability of getting from x trials until the first success.
	:param x: Number of trials until the first success.
	:param p: Probability of success.
	:param q: Probability of failure.
	:return: Returns the cumulative probability of getting from x trials until the first success.
	"""
	if q is not None:
		return 1 - gd_lt(x, p, q)
	return 1 - gd_leq(x, p)


def gd_gt(x, p, q=None):
	"""
	Computes the cumulative probability of getting from x trials until the first success.
	:param x: Number of trials until the first success.
	:param p: Probability of success.
	:param q: Probability of failure.
	:return: Returns the cumulative probability of getting from x trials until the first success.
	"""
	if q is not None:
		return 1 - gd_leq(x, p, q)
	return 1 - gd_leq(x, p)


def hgd(x, N, n, k):
	"""
	Computes the hyper geometric distribution.
	:param x: Number of successes in the sample.
	:param N: Number of items in the population.
	:param n: Number of draws.
	:param k: Number of successes in the population.
	:return: Returns the probability of getting x successes in n draws from a population of size N with k successes.
	"""
	return (combination(k, x) * combination(N - k, n - x)) / combination(N, n)


def hgd_mean(N, n, k):
	"""
	Computes the mean of the hyper geometric distribution.
	:param N: Number of items in the population.
	:param n: Number of draws.
	:param k: Number of successes in the population.
	:return: Returns the mean of the hyper geometric distribution.
	"""
	return n * (k / N)


def hgd_var(N, n, k):
	"""
	Computes the variance of the hyper geometric distribution.
	:param N: Number of items in the population.
	:param n: Number of draws.
	:param k: Number of successes in the population.
	:return: Returns the variance of the hyper geometric distribution.
	"""
	return (n * k * (N - k) * (N - n)) / (N ** 2 * (N - 1))


def hgd_std(N, n, k):
	"""
	Computes the standard deviation of the hyper geometric distribution.
	:param N: Number of items in the population.
	:param n: Number of draws.
	:param k: Number of successes in the population.
	:return: Returns the standard deviation of the hyper geometric distribution.
	"""
	return hgd_var(N, n, k) ** 0.5


def hgd_leq(x, N, n, k):
	"""
	Computes the cumulative probability of getting upto x successes in n draws from a population of size N with k successes.
	:param x: Number of successes in the sample.
	:param N: Number of items in the population.
	:param n: Number of draws.
	:param k: Number of successes in the population.
	:return: Returns the cumulative probability of getting upto x successes in n draws from a population of size N with k successes.
	"""
	return sum(hgd(i, N, n, k) for i in range(x + 1))


def hgd_lt(x, N, n, k):
	"""
	Computes the cumulative probability of getting less than x successes in n draws from a population of size N with k successes.
	:param x: Number of successes in the sample.
	:param N: Number of items in the population.
	:param n: Number of draws.
	:param k: Number of successes in the population.
	:return: Returns the cumulative probability of getting less than x successes in n draws from a population of size N with k successes.
	"""
	return sum(hgd(i, N, n, k) for i in range(x))


def hgd_geq(x, N, n, k):
	"""
	Computes the cumulative probability of getting from x successes in n draws from a population of size N with k successes.
	:param x: Number of successes in the sample.
	:param N: Number of items in the population.
	:param n: Number of draws.
	:param k: Number of successes in the population.
	:return: Returns the cumulative probability of getting from x successes in n draws from a population of size N with k successes.
	"""
	return 1 - hgd_lt(x, N, n, k)


def hgd_gt(x, N, n, k):
	"""
	Computes the cumulative probability of getting from x successes in n draws from a population of size N with k successes.
	:param x: Number of successes in the sample.
	:param N: Number of items in the population.
	:param n: Number of draws.
	:param k: Number of successes in the population.
	:return: Returns the cumulative probability of getting from x successes in n draws from a population of size N with k successes.
	"""
	return 1 - hgd_leq(x, N, n, k)


def pd(x, l):
	"""
	Computes the poisson distribution.
	:param x: Number of occurrences.
	:param l: Average number of occurrences.
	:return: Returns the probability of getting x occurrences.
	"""
	return (l ** x * math.e ** -l) / factorial(x)


def pd_mean(l):
	"""
	Computes the mean of the poisson distribution.
	:param l: Average number of occurrences.
	:return: Returns the mean of the poisson distribution.
	"""
	return l


def pd_var(l):
	"""
	Computes the variance of the poisson distribution.
	:param l: Average number of occurrences.
	:return: Returns the variance of the poisson distribution.
	"""
	return l


def pd_std(l):
	"""
	Computes the standard deviation of the poisson distribution.
	:param l: Average number of occurrences.
	:return: Returns the standard deviation of the poisson distribution.
	"""
	return l ** 0.5


def pd_leq(x, l):
	"""
	Computes the cumulative probability of getting upto x occurrences.
	:param x: Number of occurrences.
	:param l: Average number of occurrences.
	:return: Returns the cumulative probability of getting upto x occurrences.
	"""
	return sum(pd(i, l) for i in range(x + 1))


def pd_lt(x, l):
	"""
	Computes the cumulative probability of getting less than x occurrences.
	:param x: Number of occurrences.
	:param l: Average number of occurrences.
	:return: Returns the cumulative probability of getting less than x occurrences.
	"""
	return sum(pd(i, l) for i in range(x))


def pd_geq(x, l):
	"""
	Computes the cumulative probability of getting from x occurrences.
	:param x: Number of occurrences.
	:param l: Average number of occurrences.
	:return: Returns the cumulative probability of getting from x occurrences.
	"""
	return 1 - pd_lt(x, l)


def pd_gt(x, l):
	"""
	Computes the cumulative probability of getting from x occurrences.
	:param x: Number of occurrences.
	:param l: Average number of occurrences.
	:return: Returns the cumulative probability of getting from x occurrences.
	"""
	return 1 - pd_leq(x, l)


def man():
	"""
	Prints the manual for the module.
	Formatted this way to fit in memory on the calculator.
	"""
	seperator = "-" * 20
	print("This module contains functions for computing the total probability of events.")
	print("The functions are:")
	print(seperator)
	print("Binomial Distribution")
	print("bnd(x, n, p)")
	print("bnd_mean(n, p)")
	print("bnd_var(n, p)")
	print("bnd_std(n, p)")
	print("bnd_leq(x, n, p)")
	print("bnd_lt(x, n, p)")
	print("bnd_geq(x, n, p)")
	print("bnd_gt(x, n, p)")
	print(seperator)
	print("Geometric Distribution")
	print("gd(x, p, q)")
	print("gd_mean(p)")
	print("gd_var(p)")
	print("gd_std(p)")
	print("gd_leq(x, p, q)")
	print("gd_lt(x, p, q)")
	print("gd_geq(x, p, q)")
	print("gd_gt(x, p, q)")
	print(seperator)
	print("Hyper Geometric Distribution")
	print("hgd(x, N, n, k)")
	print("hgd_mean(N, n, k)")
	print("hgd_var(N, n, k)")
	print("hgd_std(N, n, k)")
	print("hgd_leq(x, N, n, k)")
	print("hgd_lt(x, N, n, k)")
	print("hgd_geq(x, N, n, k)")
	print("hgd_gt(x, N, n, k)")
	print(seperator)
	print("Poisson Distribution")
	print("pd(x, l)")
	print("pd_mean(l)")
	print("pd_var(l)")
	print("pd_std(l)")
	print("pd_leq(x, l)")
	print("pd_lt(x, l)")
	print("pd_geq(x, l)")
	print("pd_gt(x, l)")