**Source code:** Lib/random.py

This module implements pseudo-random number generators for variousdistributions.

For integers, there is uniform selection from a range. For sequences, there isuniform selection of a random element, a function to generate a randompermutation of a list in-place, and a function for random sampling withoutreplacement.

On the real line, there are functions to compute uniform, normal (Gaussian),lognormal, negative exponential, gamma, and beta distributions. For generatingdistributions of angles, the von Mises distribution is available.

Almost all module functions depend on the basic function random(), whichgenerates a random float uniformly in the half-open range `0.0 <= X < 1.0`

.Python uses the Mersenne Twister as the core generator. It produces 53-bit precisionfloats and has a period of 2**19937-1. The underlying implementation in C isboth fast and threadsafe. The Mersenne Twister is one of the most extensivelytested random number generators in existence. However, being completelydeterministic, it is not suitable for all purposes, and is completely unsuitablefor cryptographic purposes.

The functions supplied by this module are actually bound methods of a hiddeninstance of the random.Random class. You can instantiate your owninstances of Random to get generators that don’t share state.

Class Random can also be subclassed if you want to use a differentbasic generator of your own devising: see the documentation on that class formore details.

The random module also provides the SystemRandom class whichuses the system function os.urandom() to generate random numbersfrom sources provided by the operating system.

Warning

The pseudo-random generators of this module should not be used forsecurity purposes. For security or cryptographic uses, see thesecrets module.

See also

M. Matsumoto and T. Nishimura, “Mersenne Twister: A 623-dimensionallyequidistributed uniform pseudorandom number generator”, ACM Transactions onModeling and Computer Simulation Vol. 8, No. 1, January pp.3–30 1998.

Complementary-Multiply-with-Carry recipe for a compatible alternativerandom number generator with a long period and comparatively simple updateoperations.

## 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. Ifrandomness sources are provided by the operating system, they are usedinstead of the system time (see the os.urandom() function for detailson availability).If

*a*is an int, it is used directly.With version 2 (the default), a str, bytes, or bytearrayobject gets converted to an int and all of its bits are used.

With version 1 (provided for reproducing random sequences from older versionsof Python), the algorithm for str and bytes generates anarrower range of seeds.

Changed in version 3.2: Moved to the version 2 scheme which uses all of the bits in a string seed.

Changed in version 3.11: The

*seed*must be one of the following types:`None`

, int, float, str,bytes, or bytearray.

- random.getstate()¶
Return an object capturing the current internal state of the generator. Thisobject can be passed to setstate() to restore the state.

- random.setstate(
*state*)¶ *state*should have been obtained from a previous call to getstate(), andsetstate() restores the internal state of the generator to what it was atthe 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.

Added 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 roughly equivalent to

`choice(range(start, stop, step))`

butsupports arbitrarily large ranges and is optimized for common cases.The positional argument pattern matches the range() function.

Keyword arguments should not be used because they can be interpretedin unexpected ways. For example

`randrange(start=100)`

is interpretedas`randrange(0, 100, 1)`

.See Also80+ Awesome Websites for Teaching and Learning MathHow to Get 800 on SAT Math, by a Perfect ScorerSAT Practice and Preparation – SAT SuiteSec 3 A-Math & E-Math School Exam Papers. Express & Normal AcademicChanged in version 3.2: randrange() is more sophisticated about producing equally distributedvalues. Formerly it used a style like

`int(random()*n)`

which could produceslightly uneven distributions.Changed in version 3.12: Automatic conversion of non-integer types is no longer supported.Calls such as

`randrange(10.0)`

and`randrange(Fraction(10, 1))`

now raise a TypeError.

- 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 methodis supplied with the Mersenne Twister generator and some other generatorsmay also provide it as an optional part of the API. When available,getrandbits() enables randrange() to handle arbitrarily largeranges.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 therelative weights. Alternatively, if a*cum_weights*sequence is given, theselections are made according to the cumulative weights (perhaps computedusing 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 tocumulative weights before making selections, so supplying the cumulativeweights saves work.If neither

*weights*nor*cum_weights*are specified, selections are madewith equal probability. If a weights sequence is supplied, it must bethe same length as the*population*sequence. It is a TypeErrorto specify both*weights*and*cum_weights*.The

*weights*or*cum_weights*can use any numeric type that interoperateswith the float values returned by random() (that includesintegers, floats, and fractions but excludes decimals). Weights are assumedto be non-negative and finite. A ValueError is raised if allweights are zero.For a given seed, the choices() function with equal weightingtypically produces a different sequence than repeated calls tochoice(). The algorithm used by choices() uses floatingpoint arithmetic for internal consistency and speed. The algorithm usedby choice() defaults to integer arithmetic with repeated selectionsto avoid small biases from round-off error.

Added in version 3.6.

Changed in version 3.9: Raises a ValueError if all weights are zero.

- random.shuffle(
*x*)¶ Shuffle the sequence

*x*in place.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 begenerated. For example, a sequence of length 2080 is the largest thatcan fit within the period of the Mersenne Twister random number generator.Changed in version 3.11: Removed the optional parameter

*random*.

- random.sample(
*population*,*k*,***,*counts=None*)¶ Return a

*k*length list of unique elements chosen from the populationsequence. Used for random sampling without replacement.Returns a new list containing elements from the population while leaving theoriginal population unchanged. The resulting list is in selection order so thatall sub-slices will also be valid random samples. This allows raffle winners(the sample) to be partitioned into grand prize and second place winners (thesubslices).

Members of the population need not be hashable or unique. If the populationcontains repeats, then each occurrence is a possible selection in the sample.

Repeated elements can be specified one at a time or with the optionalkeyword-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 anargument. This is especially fast and space efficient for sampling from a largepopulation:

`sample(range(10000000), k=60)`

.If the sample size is larger than the population size, a ValueErroris raised.

Changed in version 3.9: Added the

*counts*parameter.Changed in version 3.11: The

*population*must be a sequence. Automatic conversion of setsto lists is no longer supported.

## Discrete distributions¶

The following function generates a discrete distribution.

- random.binomialvariate(
*n=1*,*p=0.5*)¶ Binomial distribution.Return the number of successes for

*n*independent trials with theprobability of success in each trial being*p*:Mathematically equivalent to:

sum(random() < p for i in range(n))

The number of trials

*n*should be a non-negative integer.The probability of success*p*should be between`0.0 <= p <= 1.0`

.The result is an integer in the range`0 <= X <= n`

.Added in version 3.12.

## Real-valued distributions¶

The following functions generate specific real-valued distributions. Functionparameters are named after the corresponding variables in the distribution’sequation, as used in common mathematical practice; most of these equations canbe found in any statistics text.

- random.random()¶
Return the next random floating point number in the range

`0.0 <= X < 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 rangedepending on floating-point rounding in the expression`a + (b-a) * random()`

.

- random.triangular(
*low*,*high*,*mode*)¶ Return a random floating point number

*N*such that`low <= N <= high`

andwith the specified*mode*between those bounds. The*low*and*high*boundsdefault to zero and one. The*mode*argument defaults to the midpointbetween 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=1.0*)¶ Exponential distribution.

*lambd*is 1.0 divided by the desiredmean. It should be nonzero. (The parameter would be called“lambda”, but that is a reserved word in Python.) Returned valuesrange from 0 to positive infinity if*lambd*is positive, and fromnegative infinity to 0 if*lambd*is negative.Changed in version 3.12: Added the default value for

`lambd`

.

- random.gammavariate(
*alpha*,*beta*)¶ Gamma distribution. (

*Not*the gamma function!) The shape andscale parameters,*alpha*and*beta*, must have positive values.(Calling conventions vary and some sources define ‘beta’as the inverse of the scale).The probability distribution function is:

x ** (alpha - 1) * math.exp(-x / beta)pdf(x) = -------------------------------------- math.gamma(alpha) * beta ** alpha

- random.gauss(
*mu=0.0*,*sigma=1.0*)¶ Normal distribution, also called the Gaussian distribution.

*mu*is the mean,and*sigma*is the standard deviation. This is slightly faster thanthe normalvariate() function defined below.Multithreading note: When two threads call this functionsimultaneously, it is possible that they will receive thesame return value. This can be avoided in three ways.1) Have each thread use a different instance of the randomnumber generator. 2) Put locks around all calls. 3) Use theslower, but thread-safe normalvariate() function instead.

Changed in version 3.11:

*mu*and*sigma*now have default arguments.

- random.lognormvariate(
*mu*,*sigma*)¶ Log normal distribution. If you take the natural logarithm of thisdistribution, you’ll get a normal distribution with mean

*mu*and standarddeviation*sigma*.*mu*can have any value, and*sigma*must be greater thanzero.

- random.normalvariate(
*mu=0.0*,*sigma=1.0*)¶ Normal distribution.

*mu*is the mean, and*sigma*is the standard deviation.Changed in version 3.11:

*mu*and*sigma*now have default arguments.

- 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 angleover 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 shapeparameter.

## Alternative Generator¶

*class*random.Random([*seed*])¶Class that implements the default pseudo-random number generator used by therandom module.

Changed in version 3.11: Formerly the

*seed*could be any hashable object. Now it is limited to:`None`

, int, float, str,bytes, or bytearray.Subclasses of

`Random`

should override the following methods if theywish to make use of a different basic generator:- seed(
*a=None*,*version=2*)¶ Override this method in subclasses to customise the seed()behaviour of

`Random`

instances.

- getstate()¶
Override this method in subclasses to customise the getstate()behaviour of

`Random`

instances.

- setstate(
*state*)¶ Override this method in subclasses to customise the setstate()behaviour of

`Random`

instances.

- random()¶
Override this method in subclasses to customise the random()behaviour of

`Random`

instances.

Optionally, a custom generator subclass can also supply the following method:

- getrandbits(
*k*)¶ Override this method in subclasses to customise thegetrandbits() behaviour of

`Random`

instances.

- seed(

*class*random.SystemRandom([*seed*])¶Class that uses the os.urandom() function for generating random numbersfrom 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 raiseNotImplementedError if called.

## Notes on Reproducibility¶

Sometimes it is useful to be able to reproduce the sequences given by apseudo-random number generator. By reusing a seed value, the same sequence should bereproducible from run to run as long as multiple threads are not running.

Most of the random module’s algorithms and seeding functions are subject tochange across Python versions, but two aspects are guaranteed not to change:

If a new seeding method is added, then a backward compatible seeder will beoffered.

The generator’s random() method will continue to produce the samesequence when the compatible seeder is given the same seed.

## Examples¶

Basic examples:

>>> random() # Random float: 0.0 <= x < 1.00.37444887175646646>>> uniform(2.5, 10.0) # Random float: 2.5 <= x <= 10.03.1800146073117523>>> expovariate(1 / 5) # Interval between arrivals averaging 5 seconds5.148957571865031>>> randrange(10) # Integer from 0 to 9 inclusive7>>> randrange(0, 101, 2) # Even integer from 0 to 100 inclusive26>>> 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.>>> deal = sample(['tens', 'low cards'], counts=[16, 36], k=20)>>> deal.count('tens') / 200.15>>> # Estimate the probability of getting 5 or more heads from 7 spins>>> # of a biased coin that settles on heads 60% of the time.>>> sum(binomialvariate(n=7, p=0.6) >= 5 for i in range(10_000)) / 10_0000.4169>>> # 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_0000.7958

Example of statistical bootstrapping using resamplingwith replacement to estimate a confidence interval for the mean of a sample:

# https://www.thoughtco.com/example-of-bootstrapping-3126155from statistics import fmean as meanfrom random import choicesdata = [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 testto determine the statistical significance or p-value of an observed differencebetween the effects of a drug versus a placebo:

# Example from "Statistics is Easy" by Dennis Shasha and Manda Wilsonfrom statistics import fmean as meanfrom random import shuffledrug = [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_000count = 0combined = drug + placebofor 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 heapify, heapreplacefrom random import expovariate, gaussfrom statistics import mean, quantilesaverage_arrival_interval = 5.6average_service_time = 15.0stdev_service_time = 3.5num_servers = 3waits = []arrival_time = 0.0servers = [0.0] * num_servers # time when each server becomes availableheapify(servers)for i in range(1_000_000): arrival_time += expovariate(1.0 / average_arrival_interval) next_server_available = servers[0] wait = max(0.0, next_server_available - arrival_time) waits.append(wait) service_duration = max(0.0, gauss(average_service_time, stdev_service_time)) service_completed = arrival_time + wait + service_duration heapreplace(servers, service_completed)print(f'Mean wait: {mean(waits):.1f} Max wait: {max(waits):.1f}')print('Quartiles:', [round(q, 1) for q in quantiles(waits)])

See also

Statistics for Hackersa video tutorial byJake Vanderplason statistical analysis using just a few fundamental conceptsincluding simulation, sampling, shuffling, and cross-validation.

Economics Simulationa simulation of a marketplace byPeter Norvig that shows effectiveuse 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 coveringthe basics of probability theory, how to write simulations, andhow to perform data analysis using Python.

## Recipes¶

These recipes show how to efficiently make random selectionsfrom the combinatoric iterators in the itertools module:

def random_product(*args, repeat=1): "Random selection from itertools.product(*args, **kwds)" pools = [tuple(pool) for pool in args] * repeat return tuple(map(random.choice, pools))def random_permutation(iterable, r=None): "Random selection from itertools.permutations(iterable, r)" pool = tuple(iterable) r = len(pool) if r is None else r return tuple(random.sample(pool, r))def random_combination(iterable, r): "Random selection from itertools.combinations(iterable, r)" pool = tuple(iterable) n = len(pool) indices = sorted(random.sample(range(n), r)) return tuple(pool[i] for i in indices)def random_combination_with_replacement(iterable, r): "Choose r elements with replacement. Order the result to match the iterable." # Result will be in set(itertools.combinations_with_replacement(iterable, r)). pool = tuple(iterable) n = len(pool) indices = sorted(random.choices(range(n), k=r)) return tuple(pool[i] for i in indices)

The default random() returns multiples of 2⁻⁵³ in the range*0.0 ≤ x < 1.0*. All such numbers are evenly spaced and are exactlyrepresentable as Python floats. However, many other representablefloats 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 theinterval are possible selections. The mantissa comes from a uniformdistribution of integers in the range *2⁵² ≤ mantissa < 2⁵³*. Theexponent comes from a geometric distribution where exponents smallerthan *-53* occur half as often as the next larger exponent.

from random import Randomfrom math import ldexpclass 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 distributionsin 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 fromall the multiples of 2⁻¹⁰⁷⁴ in the range *0.0 ≤ x < 1.0*. All suchnumbers are evenly spaced, but most have to be rounded down to thenearest representable Python float. (The value 2⁻¹⁰⁷⁴ is the smallestpositive unnormalized float and is equal to `math.ulp(0.0)`

.)

See also

Generating Pseudo-random Floating-Point Values apaper by Allen B. Downey describing ways to generate morefine-grained floats than normally generated by random().