random — Generate pseudo-random numbers Source code: Lib/random.py This module implements pseudo-random number generators for various distributions. For integers, there is uniform selection from a range. For sequences, there is uniform selection of a random element, a function to generate a random permutation of a list in-place, and a function for random sampling without replacement. On the real line, there are functions to compute uniform, normal (Gaussian), lognormal, negative exponential, gamma, and beta distributions. For generating distributions of angles, the von Mises distribution is available. Almost all module functions depend on the basic function random(), which generates a random float uniformly in the semi-open range [0.0, 1.0). Python uses the Mersenne Twister as the core generator. It produces 53-bit precision floats and has a period of 2**19937-1. The underlying implementation in C is both fast and threadsafe. The Mersenne Twister is one of the most extensively tested random number generators in existence. However, being completely deterministic, it is not suitable for all purposes, and is completely unsuitable for cryptographic purposes. The functions supplied by this module are actually bound methods of a hidden instance of the random.Random class. You can instantiate your own instances of Random to get generators that don’t share state. Class Random can also be subclassed if you want to use a different basic generator of your own devising: in that case, override the random(), seed(), getstate(), and setstate() methods. Optionally, a new generator can supply a getrandbits() method — this allows randrange() to produce selections over an arbitrarily large range. The random module also provides the SystemRandom class which uses the system function os.urandom() to generate random numbers from sources provided by the operating system.  Warning The pseudo-random generators of this module should not be used for security purposes. For security or cryptographic uses, see the secrets module.   See also M. Matsumoto and T. Nishimura, “Mersenne Twister: A 623-dimensionally equidistributed uniform pseudorandom number generator”, ACM Transactions on Modeling and Computer Simulation Vol. 8, No. 1, January pp.3–30 1998. Complementary-Multiply-with-Carry recipe for a compatible alternative random number generator with a long period and comparatively simple update operations.  Bookkeeping functions  
random.seed(a=None, version=2)  
Initialize the random number generator. If a is omitted or None, the current system time is used. If randomness sources are provided by the operating system, they are used instead of the system time (see the os.urandom() function for details on availability). If a is an int, it is used directly. With version 2 (the default), a str, bytes, or bytearray object gets converted to an int and all of its bits are used. With version 1 (provided for reproducing random sequences from older versions of Python), the algorithm for str and bytes generates a narrower range of seeds.  Changed in version 3.2: Moved to the version 2 scheme which uses all of the bits in a string seed.   Deprecated since version 3.9: In the future, the seed must be one of the following types: NoneType, int, float, str, bytes, or bytearray.  
  
random.getstate()  
Return an object capturing the current internal state of the generator. This object can be passed to setstate() to restore the state. 
  
random.setstate(state)  
state should have been obtained from a previous call to getstate(), and setstate() restores the internal state of the generator to what it was at the time getstate() was called. 
 Functions for bytes  
random.randbytes(n)  
Generate n random bytes. This method should not be used for generating security tokens. Use secrets.token_bytes() instead.  New in version 3.9.  
 Functions for integers  
random.randrange(stop)  
random.randrange(start, stop[, step])  
Return a randomly selected element from range(start, stop, step). This is equivalent to choice(range(start, stop, step)), but doesn’t actually build a range object. The positional argument pattern matches that of range(). Keyword arguments should not be used because the function may use them in unexpected ways.  Changed in version 3.2: randrange() is more sophisticated about producing equally distributed values. Formerly it used a style like int(random()*n) which could produce slightly uneven distributions.  
  
random.randint(a, b)  
Return a random integer N such that a <= N <= b. Alias for randrange(a, b+1). 
  
random.getrandbits(k)  
Returns a non-negative Python integer with k random bits. This method is supplied with the MersenneTwister generator and some other generators may also provide it as an optional part of the API. When available, getrandbits() enables randrange() to handle arbitrarily large ranges.  Changed in version 3.9: This method now accepts zero for k.  
 Functions for sequences  
random.choice(seq)  
Return a random element from the non-empty sequence seq. If seq is empty, raises IndexError. 
  
random.choices(population, weights=None, *, cum_weights=None, k=1)  
Return a k sized list of elements chosen from the population with replacement. If the population is empty, raises IndexError. If a weights sequence is specified, selections are made according to the relative weights. Alternatively, if a cum_weights sequence is given, the selections are made according to the cumulative weights (perhaps computed using itertools.accumulate()). For example, the relative weights [10, 5, 30, 5] are equivalent to the cumulative weights [10, 15, 45, 50]. Internally, the relative weights are converted to cumulative weights before making selections, so supplying the cumulative weights saves work. If neither weights nor cum_weights are specified, selections are made with equal probability. If a weights sequence is supplied, it must be the same length as the population sequence. It is a TypeError to specify both weights and cum_weights. The weights or cum_weights can use any numeric type that interoperates with the float values returned by random() (that includes integers, floats, and fractions but excludes decimals). Behavior is undefined if any weight is negative. A ValueError is raised if all weights are zero. For a given seed, the choices() function with equal weighting typically produces a different sequence than repeated calls to choice(). The algorithm used by choices() uses floating point arithmetic for internal consistency and speed. The algorithm used by choice() defaults to integer arithmetic with repeated selections to avoid small biases from round-off error.  New in version 3.6.   Changed in version 3.9: Raises a ValueError if all weights are zero.  
  
random.shuffle(x[, random])  
Shuffle the sequence x in place. The optional argument random is a 0-argument function returning a random float in [0.0, 1.0); by default, this is the function random(). To shuffle an immutable sequence and return a new shuffled list, use sample(x, k=len(x)) instead. Note that even for small len(x), the total number of permutations of x can quickly grow larger than the period of most random number generators. This implies that most permutations of a long sequence can never be generated. For example, a sequence of length 2080 is the largest that can fit within the period of the Mersenne Twister random number generator.  Deprecated since version 3.9, will be removed in version 3.11: The optional parameter random.  
  
random.sample(population, k, *, counts=None)  
Return a k length list of unique elements chosen from the population sequence or set. Used for random sampling without replacement. Returns a new list containing elements from the population while leaving the original population unchanged. The resulting list is in selection order so that all sub-slices will also be valid random samples. This allows raffle winners (the sample) to be partitioned into grand prize and second place winners (the subslices). Members of the population need not be hashable or unique. If the population contains repeats, then each occurrence is a possible selection in the sample. Repeated elements can be specified one at a time or with the optional keyword-only counts parameter. For example, sample(['red', 'blue'],
counts=[4, 2], k=5) is equivalent to sample(['red', 'red', 'red', 'red',
'blue', 'blue'], k=5). To choose a sample from a range of integers, use a range() object as an argument. This is especially fast and space efficient for sampling from a large population: sample(range(10000000), k=60). If the sample size is larger than the population size, a ValueError is raised.  Changed in version 3.9: Added the counts parameter.   Deprecated since version 3.9: In the future, the population must be a sequence. Instances of set are no longer supported. The set must first be converted to a list or tuple, preferably in a deterministic order so that the sample is reproducible.  
 Real-valued distributions The following functions generate specific real-valued distributions. Function parameters are named after the corresponding variables in the distribution’s equation, as used in common mathematical practice; most of these equations can be found in any statistics text.  
random.random()  
Return the next random floating point number in the range [0.0, 1.0). 
  
random.uniform(a, b)  
Return a random floating point number N such that a <= N <= b for a <= b and b <= N <= a for b < a. The end-point value b may or may not be included in the range depending on floating-point rounding in the equation a + (b-a) * random(). 
  
random.triangular(low, high, mode)  
Return a random floating point number N such that low <= N <= high and with the specified mode between those bounds. The low and high bounds default to zero and one. The mode argument defaults to the midpoint between the bounds, giving a symmetric distribution. 
  
random.betavariate(alpha, beta)  
Beta distribution. Conditions on the parameters are alpha > 0 and beta > 0. Returned values range between 0 and 1. 
  
random.expovariate(lambd)  
Exponential distribution. lambd is 1.0 divided by the desired mean. It should be nonzero. (The parameter would be called “lambda”, but that is a reserved word in Python.) Returned values range from 0 to positive infinity if lambd is positive, and from negative infinity to 0 if lambd is negative. 
  
random.gammavariate(alpha, beta)  
Gamma distribution. (Not the gamma function!) Conditions on the parameters are alpha > 0 and beta > 0. The probability distribution function is:           x ** (alpha - 1) * math.exp(-x / beta)
pdf(x) =  --------------------------------------
            math.gamma(alpha) * beta ** alpha
 
  
random.gauss(mu, sigma)  
Gaussian distribution. mu is the mean, and sigma is the standard deviation. This is slightly faster than the normalvariate() function defined below. Multithreading note: When two threads call this function simultaneously, it is possible that they will receive the same return value. This can be avoided in three ways. 1) Have each thread use a different instance of the random number generator. 2) Put locks around all calls. 3) Use the slower, but thread-safe normalvariate() function instead. 
  
random.lognormvariate(mu, sigma)  
Log normal distribution. If you take the natural logarithm of this distribution, you’ll get a normal distribution with mean mu and standard deviation sigma. mu can have any value, and sigma must be greater than zero. 
  
random.normalvariate(mu, sigma)  
Normal distribution. mu is the mean, and sigma is the standard deviation. 
  
random.vonmisesvariate(mu, kappa)  
mu is the mean angle, expressed in radians between 0 and 2*pi, and kappa is the concentration parameter, which must be greater than or equal to zero. If kappa is equal to zero, this distribution reduces to a uniform random angle over the range 0 to 2*pi. 
  
random.paretovariate(alpha)  
Pareto distribution. alpha is the shape parameter. 
  
random.weibullvariate(alpha, beta)  
Weibull distribution. alpha is the scale parameter and beta is the shape parameter. 
 Alternative Generator  
class random.Random([seed])  
Class that implements the default pseudo-random number generator used by the random module.  Deprecated since version 3.9: In the future, the seed must be one of the following types: NoneType, int, float, str, bytes, or bytearray.  
  
class random.SystemRandom([seed])  
Class that uses the os.urandom() function for generating random numbers from sources provided by the operating system. Not available on all systems. Does not rely on software state, and sequences are not reproducible. Accordingly, the seed() method has no effect and is ignored. The getstate() and setstate() methods raise NotImplementedError if called. 
 Notes on Reproducibility Sometimes it is useful to be able to reproduce the sequences given by a pseudo-random number generator. By re-using a seed value, the same sequence should be reproducible from run to run as long as multiple threads are not running. Most of the random module’s algorithms and seeding functions are subject to change across Python versions, but two aspects are guaranteed not to change:  If a new seeding method is added, then a backward compatible seeder will be offered. The generator’s random() method will continue to produce the same sequence when the compatible seeder is given the same seed.  Examples Basic examples: >>> random()                             # Random float:  0.0 <= x < 1.0
0.37444887175646646

>>> uniform(2.5, 10.0)                   # Random float:  2.5 <= x < 10.0
3.1800146073117523

>>> expovariate(1 / 5)                   # Interval between arrivals averaging 5 seconds
5.148957571865031

>>> randrange(10)                        # Integer from 0 to 9 inclusive
7

>>> randrange(0, 101, 2)                 # Even integer from 0 to 100 inclusive
26

>>> choice(['win', 'lose', 'draw'])      # Single random element from a sequence
'draw'

>>> deck = 'ace two three four'.split()
>>> shuffle(deck)                        # Shuffle a list
>>> deck
['four', 'two', 'ace', 'three']

>>> sample([10, 20, 30, 40, 50], k=4)    # Four samples without replacement
[40, 10, 50, 30]
 Simulations: >>> # Six roulette wheel spins (weighted sampling with replacement)
>>> choices(['red', 'black', 'green'], [18, 18, 2], k=6)
['red', 'green', 'black', 'black', 'red', 'black']

>>> # Deal 20 cards without replacement from a deck
>>> # of 52 playing cards, and determine the proportion of cards
>>> # with a ten-value:  ten, jack, queen, or king.
>>> dealt = sample(['tens', 'low cards'], counts=[16, 36], k=20)
>>> dealt.count('tens') / 20
0.15

>>> # Estimate the probability of getting 5 or more heads from 7 spins
>>> # of a biased coin that settles on heads 60% of the time.
>>> def trial():
...     return choices('HT', cum_weights=(0.60, 1.00), k=7).count('H') >= 5
...
>>> sum(trial() for i in range(10_000)) / 10_000
0.4169

>>> # Probability of the median of 5 samples being in middle two quartiles
>>> def trial():
...     return 2_500 <= sorted(choices(range(10_000), k=5))[2] < 7_500
...
>>> sum(trial() for i in range(10_000)) / 10_000
0.7958
 Example of statistical bootstrapping using resampling with replacement to estimate a confidence interval for the mean of a sample: # http://statistics.about.com/od/Applications/a/Example-Of-Bootstrapping.htm
from statistics import fmean as mean
from random import choices

data = [41, 50, 29, 37, 81, 30, 73, 63, 20, 35, 68, 22, 60, 31, 95]
means = sorted(mean(choices(data, k=len(data))) for i in range(100))
print(f'The sample mean of {mean(data):.1f} has a 90% confidence '
      f'interval from {means[5]:.1f} to {means[94]:.1f}')
 Example of a resampling permutation test to determine the statistical significance or p-value of an observed difference between the effects of a drug versus a placebo: # Example from "Statistics is Easy" by Dennis Shasha and Manda Wilson
from statistics import fmean as mean
from random import shuffle

drug = [54, 73, 53, 70, 73, 68, 52, 65, 65]
placebo = [54, 51, 58, 44, 55, 52, 42, 47, 58, 46]
observed_diff = mean(drug) - mean(placebo)

n = 10_000
count = 0
combined = drug + placebo
for i in range(n):
    shuffle(combined)
    new_diff = mean(combined[:len(drug)]) - mean(combined[len(drug):])
    count += (new_diff >= observed_diff)

print(f'{n} label reshufflings produced only {count} instances with a difference')
print(f'at least as extreme as the observed difference of {observed_diff:.1f}.')
print(f'The one-sided p-value of {count / n:.4f} leads us to reject the null')
print(f'hypothesis that there is no difference between the drug and the placebo.')
 Simulation of arrival times and service deliveries for a multiserver queue: from heapq import heappush, heappop
from random import expovariate, gauss
from statistics import mean, median, stdev

average_arrival_interval = 5.6
average_service_time = 15.0
stdev_service_time = 3.5
num_servers = 3

waits = []
arrival_time = 0.0
servers = [0.0] * num_servers  # time when each server becomes available
for i in range(100_000):
    arrival_time += expovariate(1.0 / average_arrival_interval)
    next_server_available = heappop(servers)
    wait = max(0.0, next_server_available - arrival_time)
    waits.append(wait)
    service_duration = gauss(average_service_time, stdev_service_time)
    service_completed = arrival_time + wait + service_duration
    heappush(servers, service_completed)

print(f'Mean wait: {mean(waits):.1f}.  Stdev wait: {stdev(waits):.1f}.')
print(f'Median wait: {median(waits):.1f}.  Max wait: {max(waits):.1f}.')
  See also Statistics for Hackers a video tutorial by Jake Vanderplas on statistical analysis using just a few fundamental concepts including simulation, sampling, shuffling, and cross-validation. Economics Simulation a simulation of a marketplace by Peter Norvig that shows effective use of many of the tools and distributions provided by this module (gauss, uniform, sample, betavariate, choice, triangular, and randrange). A Concrete Introduction to Probability (using Python) a tutorial by Peter Norvig covering the basics of probability theory, how to write simulations, and how to perform data analysis using Python.  Recipes The default random() returns multiples of 2⁻⁵³ in the range 0.0 ≤ x < 1.0. All such numbers are evenly spaced and are exactly representable as Python floats. However, many other representable floats in that interval are not possible selections. For example, 0.05954861408025609 isn’t an integer multiple of 2⁻⁵³. The following recipe takes a different approach. All floats in the interval are possible selections. The mantissa comes from a uniform distribution of integers in the range 2⁵² ≤ mantissa < 2⁵³. The exponent comes from a geometric distribution where exponents smaller than -53 occur half as often as the next larger exponent. from random import Random
from math import ldexp

class FullRandom(Random):

    def random(self):
        mantissa = 0x10_0000_0000_0000 | self.getrandbits(52)
        exponent = -53
        x = 0
        while not x:
            x = self.getrandbits(32)
            exponent += x.bit_length() - 32
        return ldexp(mantissa, exponent)
 All real valued distributions in the class will use the new method: >>> fr = FullRandom()
>>> fr.random()
0.05954861408025609
>>> fr.expovariate(0.25)
8.87925541791544
 The recipe is conceptually equivalent to an algorithm that chooses from all the multiples of 2⁻¹⁰⁷⁴ in the range 0.0 ≤ x < 1.0. All such numbers are evenly spaced, but most have to be rounded down to the nearest representable Python float. (The value 2⁻¹⁰⁷⁴ is the smallest positive unnormalized float and is equal to math.ulp(0.0).)  See also Generating Pseudo-random Floating-Point Values a paper by Allen B. Downey describing ways to generate more fine-grained floats than normally generated by random().