Enter the lower and upper limits and press "Calculate" to generate random numbers in that range. You can also set how many, the decimal places and whether repeats are allowed (everything except the limits can be left blank, which means 1 whole number, repeats allowed).
Enter the count as a whole number from 1 to 999 and the decimal places as a whole number from 0 to 50. Each press gives new random numbers, even with the same input.
Result
Enter a range in the fields on the left and press "Calculate". The random numbers will appear here.
Table of Contents
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What you can do on this page
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What is this calculation used for?
-
How to Use
-
Formula
-
Probability of each value for random whole numbers (discrete uniform distribution)
-
How likely random decimals are (probability density of the continuous uniform distribution)
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Average of random numbers (expected value of a uniform distribution)
-
Turning a random number from 0 to 1 into a random whole number in a range
-
A classic way to make pseudorandom numbers (linear congruential generator)
-
-
Symbols and terms
-
Good to know before you start
-
How to calculate it in Excel
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How to calculate it in Google Sheets
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How to calculate it in Python
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How to write it in LaTeX and other math languages (copy and paste)
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How to have ChatGPT do the calculation
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DataChef Features
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Related Features
-
NumberChef Calculators List
What you can do on this page
- Just enter a lower and an upper limit to get random numbers (numbers picked by chance) in that range on the spot
- You can make random decimals as well as whole numbers. Choose from 0 to 50 decimal places
- Generate up to 999 numbers at once, with or without repeats. When you make 2 or more, the result also shows them sorted in ascending and descending order, along with the order they were generated in
- Works for anything from a small range like 1 to 6 up to huge whole numbers with more than 20 digits
- A plain-language explanation of how random numbers work and copy-and-paste formulas for Excel, Google Sheets and Python are all on this page
The random numbers on this page are pseudorandom numbers, made by a computer with a formula (the method used is of the quality used in cryptography). They are random enough for drawings, games and simulations. To create encryption keys or passwords themselves, however, please use a dedicated tool that works offline.
What is this calculation used for?
Fair drawings, raffles and seat assignments
Give each entrant or student a number and pick with random numbers, and the choice is fair, with no one's intentions involved. Everyone has the same chance of being picked (a uniform distribution). It is a drawing whose fairness is backed by math.
The white balls in Powerball, "5 numbers from 1 to 69 with no repeats", work in exactly this way, and there are 11,238,513 possible sets of them.
Random sampling for polls and statistical surveys
How can a poll of a few thousand people tell us what the whole country thinks, without asking every American? The answer is random sampling, choosing the people to ask with random numbers. Choosing on purpose would bias the results, so picking at random is what makes a survey trustworthy.
Random digit dialing (RDD), used in many opinion polls, creates the phone numbers themselves with random numbers, so that people with unlisted numbers have the same chance of being called.
Simulation (the Monte Carlo method) in finance, weather and science
Even a problem too complex to solve with a formula can be answered approximately by running tens of thousands of random trials and adding up the results. This is called the Monte Carlo method. It is used everywhere from risk calculations at banks and ensemble weather forecasts (running many forecasts with slightly different starting conditions) to research in physics and medicine.
Its theoretical foundation is the property you learn on this page: the average of random numbers gets closer to the expected value.
Chance in games (dice, card shuffles and loot drops)
All the chance in video games, from dice rolls and card shuffles to loot drops and randomly generated maps, is made with pseudorandom numbers.
The fact that the same seed gives the same sequence is actually put to good use, in replay features and in sharing a world seed with friends so everyone plays the same map.
Random numbers behind encryption and security (and the limits of pseudorandom numbers)
Encrypted connections for online shopping and one-time passwords only work because of random numbers, numbers an attacker cannot predict.
Ordinary pseudorandom numbers could be predicted, so they cannot be used here. Instead, cryptographically secure generators and hardware random number generators, which use physical events such as heat noise, are used. This is the most demanding use of random numbers, where their quality decides how safe the world is.
Formula
Probability of each value for random whole numbers (discrete uniform distribution)
Standard notation
(the usual math form)
\(P(X=k)\)
\(=\)
\(1\)
\(\div\)
\((b-a+1)\)
In words
(symbols replaced with words)
③
\(P(X=k)\): probability of getting a given whole number \(k\)
\(=\)
①
\(1\): total probability
\(\div\)
②
\((b-a+1)\): number of possible whole numbers
The formula in words
①
Split the
\(1\): total probability
②
into equal parts by the
\((b-a+1)\): number of possible whole numbers
③
and you get the
\(P(X=k)\): probability of getting a given whole number \(k\)
(every whole number is equally likely)
Quick example
A die can be thought of as a random whole number from 1 to 6. The probability of rolling a 3 is
probability of a 3 \(P\)
\(=\)
total probability \(1\)
\(\div\)
possible values (6)
\(1 \div (6 - 1 + 1) = \frac{1}{6} \approx 0.167\ \ (16.7\%)\)
Key idea
A random number generator is built so that every number in the range is equally likely. This pattern is called a uniform distribution. With a lower limit \(a\) and an upper limit \(b\), there are \(b - a + 1\) possible whole numbers (for 1 to 6, \(6 - 1 + 1 = 6\)), so the probability of each one is 1 divided by that count. "A 3 just came up, so a 3 is less likely next time" is not true. Every draw starts fresh with the same probabilities.
How likely random decimals are (probability density of the continuous uniform distribution)
Standard notation
(the usual math form)
\(f(x)\)
\(=\)
\(1\)
\(\div\)
\((b-a)\)
In words
(symbols replaced with words)
③
\(f(x)\): probability density (height of the graph)
\(=\)
①
\(1\): total probability
\(\div\)
②
\((b-a)\): width of the range
The formula in words
①
Spread the
\(1\): total probability
②
evenly over the
\((b-a)\): width of the range
③
and you get the
\(f(x)\): probability density (height of the graph)
(the graph is a flat rectangle)
Quick example
For random decimals from 0 to 10, the height of the graph is 1 ÷ 10 = 0.1. The probability of landing between 2 and 5 is height × width
height of the graph \(f(x)\)
\(=\)
total probability \(1\)
\(\div\)
width of the range (10)
\(1 \div (10 - 0) = 0.1\)
\(0.1 \times (5 - 2) = 0.3\ \ (30\%)\)
Key idea
Random decimals have infinitely many possible values, so the probability of getting exactly 3.000… (or any other single point) is 0. That is why, for random decimals, we think about the probability of landing in a range. The probability is the area under the graph (height × width), and over the whole range it is \(0.1 \times 10 = 1\) (100%). The height \(f(x)\) is the same everywhere in the range, so every part is equally likely. That is what a uniform distribution means.
Average of random numbers (expected value of a uniform distribution)
Standard notation
(the usual math form)
\(E[X]\)
\(=\)
\((a+b)\)
\(\div\)
\(2\)
In words
(symbols replaced with words)
③
\(E[X]\): average of the random numbers (expected value)
\(=\)
①
\((a+b)\): lower limit \(a\) plus upper limit \(b\)
\(\div\)
②
\(2\): divisor (takes the middle)
The formula in words
①
Take the
\((a+b)\): lower limit \(a\) plus upper limit \(b\)
②
divide it by the
\(2\): divisor
to find the exact middle of the range
③
and you get the
\(E[X]\): average of the random numbers (expected value)
Quick example
When you make many random whole numbers from 1 to 100, their average gets closer and closer to
average (expected value) \(E[X]\)
\(=\)
lower plus upper limit (1 + 100)
\(\div\)
divisor (2)
\((1 + 100) \div 2 = 50.5\)
Key idea
A uniform distribution is symmetric within its range, so the average (expected value) is exactly in the middle. For a die (1 to 6) it is \((1+6) \div 2 = 3.5\). Each roll is a whole number, but the average of many rolls gets closer and closer to 3.5. For random decimals, the spread (variance) is \(V[X] = \frac{(b-a)^2}{12}\).
Turning a random number from 0 to 1 into a random whole number in a range
Standard notation
(the usual math form)
\(n\)
\(=\)
\(a\)
\(+\)
\(\lfloor\)
\(u\)
\(\times\)
\((b-a+1)\)
\(\rfloor\)
In words
(symbols replaced with words)
⑤
\(n\): random whole number in the range
\(=\)
④
\(a\): lower limit
\(+\)
\(\lfloor\)
①
\(u\): random number from 0 up to (not including) 1
\(\times\)
②
\((b-a+1)\): number of possible whole numbers
③
\(\rfloor\)
The formula in words
①
Take the
\(u\): random number from 0 up to (not including) 1
②
stretch it by multiplying by the
\((b-a+1)\): number of possible whole numbers
③
drop the decimal part with the floor symbol \(\lfloor\ \rfloor\) to get a whole number
④
add the
\(a\): lower limit
⑤
and you get the
\(n\): random whole number in the range
Quick example
If the random number from 0 to 1 is 0.32, turning it into a random whole number from 1 to 10 gives
random whole number \(n\)
\(=\)
lower limit (1)
\(+\)
\(\lfloor\)
random number from 0 to 1 (0.32)
\(\times\)
possible values (10)
\(\rfloor\)
\(1 + \lfloor 0.32 \times 10 \rfloor = 1 + \lfloor 3.2 \rfloor = 1 + 3 = 4\)
Key idea
Computers often make a random number first as a decimal from 0 up to (not including) 1, then use this formula to move it into the range you want. If \(u\) is spread evenly between 0 and 1, then \(n\) is equally likely to be any whole number from \(a\) to \(b\). This is exactly the formula you use to make random whole numbers from Excel's RAND function (see the Excel section).
A classic way to make pseudorandom numbers (linear congruential generator)
Standard notation
(the usual math form)
\(X_{n+1}\)
\(=\)
\((aX_n + c)\)
\(\bmod\)
\(m\)
In words
(symbols replaced with words)
③
\(X_{n+1}\): next random number
\(=\)
①
\((aX_n + c)\): current random number \(X_n\) times \(a\), plus \(c\)
\(\bmod\)
②
\(m\): divisor
The formula in words
①
Take the
\((aX_n + c)\): current random number \(X_n\) times \(a\), plus \(c\)
②
divide it by the
\(m\): divisor
and keep the remainder
③
and you get the
\(X_{n+1}\): next random number
(repeat this to make a sequence of random numbers)
Quick example
With multiplier 5, increment 3, divisor 16 and a starting value (seed) of 7, the sequence of random numbers is set as follows
next random number \(X_1\)
\(=\)
7 times 5, plus 3 (38)
\(\bmod\)
divisor (16)
\((5 \times 7 + 3) \bmod 16 = 38 \bmod 16 = 6\)
\((5 \times 6 + 3) \bmod 16 = 33 \bmod 16 = 1\)
Key idea
Just repeating a simple step, "multiply, add, keep the remainder", produces a sequence of numbers that looks random. That is the basic idea behind pseudorandom numbers. (Here \(a, c, m\) are constants chosen in advance. They are not the lower and upper limits from the formulas above.)
The key property is that the same starting value (seed) always gives exactly the same sequence. In other words, random numbers made by a formula are not truly random. Real computers use improved methods such as the Mersenne Twister instead of a plain linear congruential generator, but the core idea, "a sequence of calculations decided by the seed", is the same.
The numbers from a random number generator follow a uniform distribution, where every number in the range is equally likely. The most important point is that computer random numbers are pseudorandom numbers made by a formula. The same seed (starting value) always gives the same sequence, so they are not true chance.
Symbols and terms
Symbols
| \(a\) | a | The lower limit of the range (this number or more). (Example - for random numbers from 1 to 100, \(a = 1\)) |
| \(b\) | b | The upper limit of the range (this number or less). (Example - for random numbers from 1 to 100, \(b = 100\)) |
| \(P(X=k)\) | P of X equals k | The probability that the random number \(X\) is exactly the whole number \(k\). For uniform random whole numbers, it is the same for every \(k\). |
| \(f(x)\) | f of x (probability density) | The height of the graph that shows how likely random decimals are (the probability density function). For a uniform distribution, the height is the same everywhere in the range. |
| \(E[X]\) | E of X (expected value) | The value that the average gets closer to when you generate many random numbers \(X\). E stands for "expectation". |
| \(u\) | u | A uniform random number from 0 up to (not including) 1. Most computer random numbers are first made in this form, then moved into the range you want. |
| \(\lfloor\ \rfloor\) | floor (floor function) | A symbol that drops the decimal part of the number inside and rounds it down to a whole number. (Example - \(\lfloor 3.2 \rfloor = 3\)) |
| \(X_n\) | X sub n | The \(n\)th value in a sequence of pseudorandom numbers. The starting value \(X_0\) is the seed. |
| \(\bmod\) | mod | The operation that gives the remainder of a division. (Example - \(38 \bmod 16 = 6\), because 38 divided by 16 is 2 with remainder 6) |
Terms
| random number | A number with no pattern that lets you predict what comes next, like the roll of a die. The generator on this page makes numbers picked at random from the range you set. |
| uniform distribution | A distribution where every value in the range is equally likely. For random whole numbers, every whole number has the same probability. For random decimals, the graph has the same height everywhere. |
| pseudorandom number | A number from a sequence made by a formula that looks random. The formula is called a PRNG (pseudorandom number generator). Almost all computer random numbers are pseudorandom numbers. |
| seed | The starting value for calculating pseudorandom numbers. Starting from the same seed always gives exactly the same sequence. Normally the seed is set automatically, for example from the current time, so each sequence looks different. |
| linear congruential generator | The most classic way to make pseudorandom numbers, by repeating "multiply, add, keep the remainder". It is ideal for learning how they work, but today higher-quality methods such as the Mersenne Twister are the norm in practice. |
| hardware random number generator | A device that makes random numbers from physical events, such as dice, coins, electrical noise in the atmosphere or heat noise in electronic circuits. Since no formula is involved, the numbers cannot be reproduced and come close to true random numbers. |
| expected value | The average of a chance outcome over many repeats. For uniform random numbers, the expected value is exactly the middle of the range. |
| random sampling | Choosing who or what to study at random, for example with random numbers, with no human choice involved. It is the basic way to avoid bias in statistical surveys. |
Good to know before you start
Here is what helps you use the calculation on this page with real understanding, not just by pressing the button.
| Division with remainders (Grade 4) |
|
| Decimals and rounding down (Grades 4–5) |
|
| Ratios and percents (Grade 6) |
|
| Basic probability (Grade 7) |
|
| Expected value (high school, Statistics) |
|
How to calculate it in Excel
Copy the whole table below and paste it into cell A1 in Excel. It works as is.
Table to find the probability of each whole number
| Lower limit a | 1 |
| Upper limit b | 6 |
| Probability of each whole number | =1/(B2-B1+1) |
Table to find the graph height (probability density) for random decimals
| Lower limit a | 0 |
| Upper limit b | 10 |
| Probability density (graph height) | =1/(B2-B1) |
Table to find the average (expected value) of the random numbers
| Lower limit a | 1 |
| Upper limit b | 100 |
| Average (expected value) | =(B1+B2)/2 |
Table to make a random whole number
| Lower limit a | 1 |
| Upper limit b | 100 |
| Random whole number (changes on every recalculation) | =B1+INT(RAND()*(B2-B1+1)) |
Table to find the next random number with a linear congruential generator
| Current random number Xn | 7 |
| Multiplier a | 5 |
| Increment c | 3 |
| Divisor m | 16 |
| Next random number | =MOD(B2*B1+B3,B4) |
After pasting, the upper rows are your inputs and the formula in the last row is calculated automatically.
The first table shows about 0.1667 (= 1/6), the third shows 50.5, and the fifth shows 6.
RAND() in the fourth table makes a random number from 0 up to (not including) 1, in exactly the form of the conversion formula in the Formula section. Press F9 to get a new random number each time.
Excel also has the function =RANDBETWEEN(B1,B2), which gives the same kind of random whole number with a single function.
The first table shows about 0.1667 (= 1/6), the third shows 50.5, and the fifth shows 6.
RAND() in the fourth table makes a random number from 0 up to (not including) 1, in exactly the form of the conversion formula in the Formula section. Press F9 to get a new random number each time.
Excel also has the function =RANDBETWEEN(B1,B2), which gives the same kind of random whole number with a single function.
How to calculate it in Google Sheets
Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.
Table to find the probability of each whole number
| Lower limit a | 1 |
| Upper limit b | 6 |
| Probability of each whole number | =1/(B2-B1+1) |
Table to find the graph height (probability density) for random decimals
| Lower limit a | 0 |
| Upper limit b | 10 |
| Probability density (graph height) | =1/(B2-B1) |
Table to find the average (expected value) of the random numbers
| Lower limit a | 1 |
| Upper limit b | 100 |
| Average (expected value) | =(B1+B2)/2 |
Table to make a random whole number
| Lower limit a | 1 |
| Upper limit b | 100 |
| Random whole number (changes on every recalculation) | =B1+INT(RAND()*(B2-B1+1)) |
Table to find the next random number with a linear congruential generator
| Current random number Xn | 7 |
| Multiplier a | 5 |
| Increment c | 3 |
| Divisor m | 16 |
| Next random number | =MOD(B2*B1+B3,B4) |
The same formulas as in Excel work as is (RAND, RANDBETWEEN, INT and MOD have the same names in Google Sheets).
Copy the whole table, paste it into cell A1, and replace the input numbers with your own.
Copy the whole table, paste it into cell A1, and replace the input numbers with your own.
How to calculate it in Python
import random
lower = 1 # lower limit
upper = 100 # upper limit
count = 5 # how many to generate
# random whole numbers (repeats allowed)
integers = [random.randint(lower, upper) for _ in range(count)]
# random whole numbers (no repeats)
unique_integers = random.sample(range(lower, upper + 1), count)
# random decimals
decimals = [random.uniform(lower, upper) for _ in range(count)]
print(f"Random whole numbers (repeats allowed): {integers}")
print(f"Random whole numbers (no repeats): {unique_integers}")
print(f"Random decimals: {decimals}")
# with a fixed seed, the same sequence comes back every time you run it (a property of pseudorandom numbers)
random.seed(42)
print(f"Random numbers with seed 42: {[random.randint(1, 100) for _ in range(3)]}")
Runs with just the standard random module (a pseudorandom number generator called the Mersenne Twister). Run the last two lines and you can see that the same seed gives the same result every time. To generate passwords or encryption keys, use the dedicated secrets module instead of random.
How to write it in LaTeX and other math languages (copy and paste)
Probability of each value for random whole numbers (discrete uniform distribution)
P(X = k) = 1 / (b − a + 1)
P(X = k) = \frac{1}{b - a + 1}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>P</mi><mo>(</mo><mi>X</mi><mo>=</mo><mi>k</mi><mo>)</mo>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow><mi>b</mi><mo>−</mo><mi>a</mi><mo>+</mo><mn>1</mn></mrow>
</mfrac>
</mrow>
</math>
P(X = k) = 1/(b - a + 1)
1/(b - a + 1)
p := 1/(b - a + 1);
p = 1/(b - a + 1);
P(X = k) = 1/(b - a + 1)
How likely random decimals are (probability density of the continuous uniform distribution)
f(x) = 1 / (b − a)
f(x) = \frac{1}{b - a}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow><mi>b</mi><mo>−</mo><mi>a</mi></mrow>
</mfrac>
</mrow>
</math>
f(x) = 1/(b - a)
1/(b - a)
f := 1/(b - a);
f = 1/(b - a);
f(x) = 1/(b - a)
Average of random numbers (expected value of a uniform distribution)
E[X] = (a + b) / 2
E[X] = \frac{a + b}{2}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>E</mi><mo>[</mo><mi>X</mi><mo>]</mo>
<mo>=</mo>
<mfrac>
<mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow>
<mn>2</mn>
</mfrac>
</mrow>
</math>
E[X] = (a + b)/2
(a + b)/2
mean := (a + b)/2;
mean_x = (a + b)/2;
E[X] = (a + b)/2
Turning a random number from 0 to 1 into a random whole number in a range
n = a + ⌊u × (b − a + 1)⌋
n = a + \lfloor u \times (b - a + 1) \rfloor
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>n</mi>
<mo>=</mo>
<mi>a</mi>
<mo>+</mo>
<mo>⌊</mo>
<mi>u</mi>
<mo>×</mo>
<mo>(</mo><mi>b</mi><mo>−</mo><mi>a</mi><mo>+</mo><mn>1</mn><mo>)</mo>
<mo>⌋</mo>
</mrow>
</math>
n = a + |__ u xx (b - a + 1) __|
a + Floor[u*(b - a + 1)]
n := a + floor(u*(b - a + 1));
n = a + floor(u*(b - a + 1));
n = a + ⌊u × (b - a + 1)⌋
A classic way to make pseudorandom numbers (linear congruential generator)
Xₙ₊₁ = (a·Xₙ + c) mod m
X_{n+1} = (a X_n + c) \bmod m
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<msub><mi>X</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub>
<mo>=</mo>
<mo>(</mo>
<mi>a</mi><msub><mi>X</mi><mi>n</mi></msub>
<mo>+</mo><mi>c</mi>
<mo>)</mo>
<mo>mod</mo>
<mi>m</mi>
</mrow>
</math>
X_(n+1) = (a X_n + c) mod m
Mod[a*x + c, m]
X[n + 1] := (a*X[n] + c) mod m;
x_next = mod(a*x + c, m);
X_(n+1) = (a X_n + c) mod m
How to have ChatGPT do the calculation
You are an assistant for generating random numbers. Do the following by actually running Python code, and base your answer only on the numbers from the execution result (random numbers cannot be made without running code, so do not guess or make up any numbers). 1. Generate 5 random whole numbers from 1 to 100 with no repeats. 2. Sort those 5 numbers in ascending order. 3. Generate 1 random decimal from 0.2 to 112.5 with 3 decimal places. Show the code you used and the numbers from the execution result.
How to Use
-
1Enter your numbersType the numbers you want to calculate with into the input fields
-
2CalculatePress the "Calculate" button
-
3Check the resultThe result appears on the spot. The same page also explains the idea behind the calculation and the formula
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