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Javascript smart and good way to make array of size N where each object in the array has a certain probability? [duplicate]

I'm trying to devise a (good) way to choose a random number from a range of possible numbers where each number in the range is given a weight. To put it simply: given the range of numbers (0,1,2) choose a number where 0 has an 80% probability of being selected, 1 has a 10% chance and 2 has a 10% chance.
It's been about 8 years since my college stats class, so you can imagine the proper formula for this escapes me at the moment.
Here's the 'cheap and dirty' method that I came up with. This solution uses ColdFusion. Yours may use whatever language you'd like. I'm a programmer, I think I can handle porting it. Ultimately my solution needs to be in Groovy - I wrote this one in ColdFusion because it's easy to quickly write/test in CF.
public function weightedRandom( Struct options ) {
var tempArr = [];
for( var o in arguments.options )
{
var weight = arguments.options[ o ] * 10;
for ( var i = 1; i<= weight; i++ )
{
arrayAppend( tempArr, o );
}
}
return tempArr[ randRange( 1, arrayLen( tempArr ) ) ];
}
// test it
opts = { 0=.8, 1=.1, 2=.1 };
for( x = 1; x<=10; x++ )
{
writeDump( weightedRandom( opts ) );
}
I'm looking for better solutions, please suggest improvements or alternatives.

Rejection sampling (such as in your solution) is the first thing that comes to mind, whereby you build a lookup table with elements populated by their weight distribution, then pick a random location in the table and return it. As an implementation choice, I would make a higher order function which takes a spec and returns a function which returns values based on the distribution in the spec, this way you avoid having to build the table for each call. The downsides are that the algorithmic performance of building the table is linear by the number of items and there could potentially be a lot of memory usage for large specs (or those with members with very small or precise weights, e.g. {0:0.99999, 1:0.00001}). The upside is that picking a value has constant time, which might be desirable if performance is critical. In JavaScript:
function weightedRand(spec) {
var i, j, table=[];
for (i in spec) {
// The constant 10 below should be computed based on the
// weights in the spec for a correct and optimal table size.
// E.g. the spec {0:0.999, 1:0.001} will break this impl.
for (j=0; j<spec[i]*10; j++) {
table.push(i);
}
}
return function() {
return table[Math.floor(Math.random() * table.length)];
}
}
var rand012 = weightedRand({0:0.8, 1:0.1, 2:0.1});
rand012(); // random in distribution...
Another strategy is to pick a random number in [0,1) and iterate over the weight specification summing the weights, if the random number is less than the sum then return the associated value. Of course, this assumes that the weights sum to one. This solution has no up-front costs but has average algorithmic performance linear by the number of entries in the spec. For example, in JavaScript:
function weightedRand2(spec) {
var i, sum=0, r=Math.random();
for (i in spec) {
sum += spec[i];
if (r <= sum) return i;
}
}
weightedRand2({0:0.8, 1:0.1, 2:0.1}); // random in distribution...

Generate a random number R between 0 and 1.
If R in [0, 0.1) -> 1
If R in [0.1, 0.2) -> 2
If R in [0.2, 1] -> 3
If you can't directly get a number between 0 and 1, generate a number in a range that will produce as much precision as you want. For example, if you have the weights for
(1, 83.7%) and (2, 16.3%), roll a number from 1 to 1000. 1-837 is a 1. 838-1000 is 2.

I use the following
function weightedRandom(min, max) {
return Math.round(max / (Math.random() * max + min));
}
This is my go-to "weighted" random, where I use an inverse function of "x" (where x is a random between min and max) to generate a weighted result, where the minimum is the most heavy element, and the maximum the lightest (least chances of getting the result)
So basically, using weightedRandom(1, 5) means the chances of getting a 1 are higher than a 2 which are higher than a 3, which are higher than a 4, which are higher than a 5.
Might not be useful for your use case but probably useful for people googling this same question.
After a 100 iterations try, it gave me:
==================
| Result | Times |
==================
| 1 | 55 |
| 2 | 28 |
| 3 | 8 |
| 4 | 7 |
| 5 | 2 |
==================

Here are 3 solutions in javascript since I'm not sure which language you want it in. Depending on your needs one of the first two might work, but the the third one is probably the easiest to implement with large sets of numbers.
function randomSimple(){
return [0,0,0,0,0,0,0,0,1,2][Math.floor(Math.random()*10)];
}
function randomCase(){
var n=Math.floor(Math.random()*100)
switch(n){
case n<80:
return 0;
case n<90:
return 1;
case n<100:
return 2;
}
}
function randomLoop(weight,num){
var n=Math.floor(Math.random()*100),amt=0;
for(var i=0;i<weight.length;i++){
//amt+=weight[i]; *alternative method
//if(n<amt){
if(n<weight[i]){
return num[i];
}
}
}
weight=[80,90,100];
//weight=[80,10,10]; *alternative method
num=[0,1,2]

8 years late but here's my solution in 4 lines.
Prepare an array of probability mass function such that
pmf[array_index] = P(X=array_index):
var pmf = [0.8, 0.1, 0.1]
Prepare an array for the corresponding cumulative distribution function such that
cdf[array_index] = F(X=array_index):
var cdf = pmf.map((sum => value => sum += value)(0))
// [0.8, 0.9, 1]
3a) Generate a random number.
3b) Get an array of elements that are more than or equal to this number.
3c) Return its length.
var r = Math.random()
cdf.filter(el => r >= el).length

This is more or less a generic-ized version of what #trinithis wrote, in Java: I did it with ints rather than floats to avoid messy rounding errors.
static class Weighting {
int value;
int weighting;
public Weighting(int v, int w) {
this.value = v;
this.weighting = w;
}
}
public static int weightedRandom(List<Weighting> weightingOptions) {
//determine sum of all weightings
int total = 0;
for (Weighting w : weightingOptions) {
total += w.weighting;
}
//select a random value between 0 and our total
int random = new Random().nextInt(total);
//loop thru our weightings until we arrive at the correct one
int current = 0;
for (Weighting w : weightingOptions) {
current += w.weighting;
if (random < current)
return w.value;
}
//shouldn't happen.
return -1;
}
public static void main(String[] args) {
List<Weighting> weightings = new ArrayList<Weighting>();
weightings.add(new Weighting(0, 8));
weightings.add(new Weighting(1, 1));
weightings.add(new Weighting(2, 1));
for (int i = 0; i < 100; i++) {
System.out.println(weightedRandom(weightings));
}
}

How about
int [ ] numbers = { 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 2 } ;
then you can randomly select from numbers and 0 will have an 80% chance, 1 10%, and 2 10%

This one is in Mathematica, but it's easy to copy to another language, I use it in my games and it can handle decimal weights:
weights = {0.5,1,2}; // The weights
weights = N#weights/Total#weights // Normalize weights so that the list's sum is always 1.
min = 0; // First min value should be 0
max = weights[[1]]; // First max value should be the first element of the newly created weights list. Note that in Mathematica the first element has index of 1, not 0.
random = RandomReal[]; // Generate a random float from 0 to 1;
For[i = 1, i <= Length#weights, i++,
If[random >= min && random < max,
Print["Chosen index number: " <> ToString#i]
];
min += weights[[i]];
If[i == Length#weights,
max = 1,
max += weights[[i + 1]]
]
]
(Now I'm talking with a lists first element's index equals 0) The idea behind this is that having a normalized list weights there is a chance of weights[n] to return the index n, so the distances between the min and max at step n should be weights[n]. The total distance from the minimum min (which we put it to be 0) and the maximum max is the sum of the list weights.
The good thing behind this is that you don't append to any array or nest for loops, and that increases heavily the execution time.
Here is the code in C# without needing to normalize the weights list and deleting some code:
int WeightedRandom(List<float> weights) {
float total = 0f;
foreach (float weight in weights) {
total += weight;
}
float max = weights [0],
random = Random.Range(0f, total);
for (int index = 0; index < weights.Count; index++) {
if (random < max) {
return index;
} else if (index == weights.Count - 1) {
return weights.Count-1;
}
max += weights[index+1];
}
return -1;
}

I suggest to use a continuous check of the probability and the rest of the random number.
This function sets first the return value to the last possible index and iterates until the rest of the random value is smaller than the actual probability.
The probabilities have to sum to one.
function getRandomIndexByProbability(probabilities) {
var r = Math.random(),
index = probabilities.length - 1;
probabilities.some(function (probability, i) {
if (r < probability) {
index = i;
return true;
}
r -= probability;
});
return index;
}
var i,
probabilities = [0.8, 0.1, 0.1],
count = probabilities.map(function () { return 0; });
for (i = 0; i < 1e6; i++) {
count[getRandomIndexByProbability(probabilities)]++;
}
console.log(count);
.as-console-wrapper { max-height: 100% !important; top: 0; }

Thanks all, this was a helpful thread. I encapsulated it into a convenience function (Typescript). Tests below (sinon, jest). Could definitely be a bit tighter, but hopefully it's readable.
export type WeightedOptions = {
[option: string]: number;
};
// Pass in an object like { a: 10, b: 4, c: 400 } and it'll return either "a", "b", or "c", factoring in their respective
// weight. So in this example, "c" is likely to be returned 400 times out of 414
export const getRandomWeightedValue = (options: WeightedOptions) => {
const keys = Object.keys(options);
const totalSum = keys.reduce((acc, item) => acc + options[item], 0);
let runningTotal = 0;
const cumulativeValues = keys.map((key) => {
const relativeValue = options[key]/totalSum;
const cv = {
key,
value: relativeValue + runningTotal
};
runningTotal += relativeValue;
return cv;
});
const r = Math.random();
return cumulativeValues.find(({ key, value }) => r <= value)!.key;
};
Tests:
describe('getRandomWeightedValue', () => {
// Out of 1, the relative and cumulative values for these are:
// a: 0.1666 -> 0.16666
// b: 0.3333 -> 0.5
// c: 0.5 -> 1
const values = { a: 10, b: 20, c: 30 };
it('returns appropriate values for particular random value', () => {
// any random number under 0.166666 should return "a"
const stub1 = sinon.stub(Math, 'random').returns(0);
const result1 = randomUtils.getRandomWeightedValue(values);
expect(result1).toEqual('a');
stub1.restore();
const stub2 = sinon.stub(Math, 'random').returns(0.1666);
const result2 = randomUtils.getRandomWeightedValue(values);
expect(result2).toEqual('a');
stub2.restore();
// any random number between 0.166666 and 0.5 should return "b"
const stub3 = sinon.stub(Math, 'random').returns(0.17);
const result3 = randomUtils.getRandomWeightedValue(values);
expect(result3).toEqual('b');
stub3.restore();
const stub4 = sinon.stub(Math, 'random').returns(0.3333);
const result4 = randomUtils.getRandomWeightedValue(values);
expect(result4).toEqual('b');
stub4.restore();
const stub5 = sinon.stub(Math, 'random').returns(0.5);
const result5 = randomUtils.getRandomWeightedValue(values);
expect(result5).toEqual('b');
stub5.restore();
// any random number above 0.5 should return "c"
const stub6 = sinon.stub(Math, 'random').returns(0.500001);
const result6 = randomUtils.getRandomWeightedValue(values);
expect(result6).toEqual('c');
stub6.restore();
const stub7 = sinon.stub(Math, 'random').returns(1);
const result7 = randomUtils.getRandomWeightedValue(values);
expect(result7).toEqual('c');
stub7.restore();
});
});

Shortest solution in modern JavaScript
Note: all weights need to be integers
function weightedRandom(items){
let table = Object.entries(items)
.flatMap(([item, weight]) => Array(item).fill(weight))
return table[Math.floor(Math.random() * table.length)]
}
const key = weightedRandom({
"key1": 1,
"key2": 4,
"key3": 8
}) // returns e.g. "key1"

here is the input and ratios : 0 (80%), 1(10%) , 2 (10%)
lets draw them out so its easy to visualize.
0 1 2
-------------------------------------________+++++++++
lets add up the total weight and call it TR for total ratio. so in this case 100.
lets randomly get a number from (0-TR) or (0 to 100 in this case) . 100 being your weights total. Call it RN for random number.
so now we have TR as the total weight and RN as the random number between 0 and TR.
so lets imagine we picked a random # from 0 to 100. Say 21. so thats actually 21%.
WE MUST CONVERT/MATCH THIS TO OUR INPUT NUMBERS BUT HOW ?
lets loop over each weight (80, 10, 10) and keep the sum of the weights we already visit.
the moment the sum of the weights we are looping over is greater then the random number RN (21 in this case), we stop the loop & return that element position.
double sum = 0;
int position = -1;
for(double weight : weight){
position ++;
sum = sum + weight;
if(sum > 21) //(80 > 21) so break on first pass
break;
}
//position will be 0 so we return array[0]--> 0
lets say the random number (between 0 and 100) is 83. Lets do it again:
double sum = 0;
int position = -1;
for(double weight : weight){
position ++;
sum = sum + weight;
if(sum > 83) //(90 > 83) so break
break;
}
//we did two passes in the loop so position is 1 so we return array[1]---> 1

I have a slotmachine and I used the code below to generate random numbers. In probabilitiesSlotMachine the keys are the output in the slotmachine, and the values represent the weight.
const probabilitiesSlotMachine = [{0 : 1000}, {1 : 100}, {2 : 50}, {3 : 30}, {4 : 20}, {5 : 10}, {6 : 5}, {7 : 4}, {8 : 2}, {9 : 1}]
var allSlotMachineResults = []
probabilitiesSlotMachine.forEach(function(obj, index){
for (var key in obj){
for (var loop = 0; loop < obj[key]; loop ++){
allSlotMachineResults.push(key)
}
}
});
Now to generate a random output, I use this code:
const random = allSlotMachineResults[Math.floor(Math.random() * allSlotMachineResults.length)]

Enjoy the O(1) (constant time) solution for your problem.
If the input array is small, it can be easily implemented.
const number = Math.floor(Math.random() * 99); // Generate a random number from 0 to 99
let element;
if (number >= 0 && number <= 79) {
/*
In the range of 0 to 99, every number has equal probability
of occurring. Therefore, if you gather 80 numbers (0 to 79) and
make a "sub-group" of them, then their probabilities will get added.
Hence, what you get is an 80% chance that the number will fall in this
range.
So, quite naturally, there is 80% probability that this code will run.
Now, manually choose / assign element of your array to this variable.
*/
element = 0;
}
else if (number >= 80 && number <= 89) {
// 10% chance that this code runs.
element = 1;
}
else if (number >= 90 && number <= 99) {
// 10% chance that this code runs.
element = 2;
}

How to divide number n in javascript into x parts, where the sum of all the parts equals the number?

I have a number which I need to divide into 5 parts. However, I want each part to be a random number. But when all the parts are added together, they equal the original number. I am unsure of how to do this with JavaScript. Furthermore, I don't want the min of the divided parts to be 0 or 1, I want to set the min myself.
For example, the number is 450. I want the divided parts to be no less than 60. So to start, the array would be [60,60,60,60,60]. But I want to randomize so that they all add up to 450. What would be the best way to go about doing this?
Thank you!
This is what I've tried so far:
let i = 0;
let number = 450;
let numArray = [];
while(i <= 5){
while(number > 0) {
let randomNum = Math.round(Math.random() * number) + 1;
numArray.push(randomNum);
number -= randomNum;
}
i += 1;
}

let your number be N, and let pn be the nth part. To get 5 parts:
p1 = random number between 0 and N
p2 = random number between 0 and N - p1
p3 = random number between 0 and N - p2 - p1
p4 = random number between 0 and N - p3 - p2 - p1
p5 = N - p4 - p3 - p2 - p1
Edit 2017
To make it seem more random, shuffle the numbers after you generate them
Edit 2020
I guess some code wouldn't hurt. Using ES7 generators:
function* splitNParts(num, parts) {
let sumParts = 0;
for (let i = 0; i < parts - 1; i++) {
const pn = Math.ceil(Math.random() * (num - sumParts))
yield pn
sumParts += pn
}
yield num - sumParts;
}
Fiddle Link

Sum the five minimums (eg min = 60) up:
var minSum = 5 * min
Then get the difference between your original number (orNumber = 450) and minSum.
var delta = orNumber - minSum
Now you get 4 different random numbers in the range from 0 to exclusive 1.
Sort these numbers ascending.
Foreach of these randoms do the following:
Subtract it from the last one (or zero for the first)
Multiply this number with the delta and you get one of the parts.
The last part is the delta minus all other parts.
Afterwards you just have to add your min to all of the parts.

This function generates random numbers from 0 to 1, adds them together to figure out what they need to be multiplied by to provide the correct range. It has the benefit of all the numbers being fairly distributed.
function divvy(number, parts, min) {
var randombit = number - min * parts;
var out = [];
for (var i=0; i < parts; i++) {
out.push(Math.random());
}
var mult = randombit / out.reduce(function (a,b) {return a+b;});
return out.map(function (el) { return el * mult + min; });
}
var d = divvy(450, 6, 60)
console.log(d);
console.log("sum - " + d.reduce(function(a,b){return a+b}));

You can use a do..while loop to subtract a minimum number from original number, keep a copy of original number for subtraction at conclusion of loop to push the remainder to the array
let [n, total, m = n] = [450, 0];
const [min, arr] = [60, []];
do {
n -= min; // subtract `min` from `n`
arr.push(n > min ? min : m - total); // push `min` or remainder
total += arr[arr.length - 1]; // keep track of total
} while (n > min);
console.log(arr);
To randomize output at resulting array select a number greater than min and less than n to create a random number within a specific range
let [n, total, m = n] = [450, 0];
const [min, arr, range = min + min / 2] = [60, []];
do {
let r = Math.random() * (range - min) + min; // random number in our range
n -= r; // subtract `min` from `n`
arr.push(n > min ? r : m - total); // push `r` or remainder
total += arr[arr.length - 1]; // keep track of total
} while (n > min);
console.log(arr);

I made a longer version for beginners.
const n = 450;
const iterations = 5;
const parts = [];
// we'll use this to store what's left on each iteration
let remainder = n;
for (let i = 1; i <= iterations; i += 1) {
// if it's the last iteration, we should just use whatever
// is left after removing all the other random numbers
// from our 450
if (i === iterations) {
parts.push(remainder);
break;
}
// every time we loop, a random number is created.
// on the first iteration, the remainder is still 450
const part = Math.round(Math.random() * remainder);
parts.push(part);
// we must store how much is left after our random numbers
// are deducted from our 450. we will use the lower number
// to calculate the next random number
remainder -= part;
}
// let's print out the array and the proof it still adds up
const total = totalFromParts(parts);
console.log(parts);
console.log('Total is still ' + total);
// this function loops through each array item, and adds it to the last
// just here to test the result
function totalFromParts(parts) {
return parts.reduce((sum, value) => sum + value, 0);
}

There are much more efficient ways to code this, but in the interest of explaining the logic of solving the problem, this walks through that step by step, transforming the values and explaining the logic.
// Set start number, number of fragments
// minimum fragment size, define fragments array
var n = 450
var x = 5
var minNumber = 60
var fragment = n / x
// stuff array with equal sized fragment values
var fragments = []
for (i = 0; i < x; i++) {
fragments[i] = fragment;
}
document.write("fragments: " + fragments);
var delta = [];
// iterate through fragments array
// get a random number each time between the fragment size
// and the minimum fragment sized defined above
// for even array slots, subtract the value from the fragment
// for odd array slots, add the value to the fragment
// skip the first [0] value
for (i = 1; i< x; i++) {
delta[i] = Math.floor(Math.random() * (fragment - minNumber));
document.write("<br />delta: " + delta[i]);
if((i % 2) == 1) {
fragments[i] -= delta[i]
}
else {
fragments[i] += delta[i]
}
}
// set the initial fragment value to 0
fragments[0] = 0
// defines a function we can use to total the array values
function getSum(total, num) {
return total + num;
}
// get the total of the array values, remembering the first is 0
var partialTotal = fragments.reduce(getSum)
document.write("<br />partial sum: " + partialTotal);
// set the first array value to the difference between
// the total of all the other array values and the original
// number the array was to sum up to
fragments[0] = (n - partialTotal)
// write the values out and profit.
document.write("<br />fragments: " + fragments);
var grandTotal = fragments.reduce(getSum)
document.write("<br />Grand total: " + grandTotal);
https://plnkr.co/edit/oToZe7LGpQS4dIVgYHPi?p=preview

Get a random number focused on center

Is it possible to get a random number between 1-100 and keep the results mainly within the 40-60 range? I mean, it will go out of that range rarely, but I want it to be mainly within that range... Is it possible with JavaScript/jQuery?
Right now I'm just using the basic Math.random() * 100 + 1.

The simplest way would be to generate two random numbers from 0-50 and add them together.
This gives a distribution biased towards 50, in the same way rolling two dice biases towards 7.
In fact, by using a larger number of "dice" (as #Falco suggests), you can make a closer approximation to a bell-curve:
function weightedRandom(max, numDice) {
let num = 0;
for (let i = 0; i < numDice; i++) {
num += Math.random() * (max/numDice);
}
return num;
}
JSFiddle: http://jsfiddle.net/797qhcza/1/

You have some good answers here that give specific solutions; let me describe for you the general solution. The problem is:
I have a source of more-or-less uniformly distributed random numbers between 0 and 1.
I wish to produce a sequence of random numbers that follow a different distribution.
The general solution to this problem is to work out the quantile function of your desired distribution, and then apply the quantile function to the output of your uniform source.
The quantile function is the inverse of the integral of your desired distribution function. The distribution function is the function where the area under a portion of the curve is equal to the probability that the randomly-chosen item will be in that portion.
I give an example of how to do so here:
http://ericlippert.com/2012/02/21/generating-random-non-uniform-data/
The code in there is in C#, but the principles apply to any language; it should be straightforward to adapt the solution to JavaScript.

Taking arrays of numbers, etc. isn't efficient. You should take a mapping which takes a random number between 0 to 100 and maps to the distribution you need. So in your case, you could take f(x)=-(1/25)x2+4x to get a distribution with the most values in the middle of your range.

I might do something like setup a "chance" for the number to be allowed to go "out of bounds". In this example, a 20% chance the number will be 1-100, otherwise, 40-60:
$(function () {
$('button').click(function () {
var outOfBoundsChance = .2;
var num = 0;
if (Math.random() <= outOfBoundsChance) {
num = getRandomInt(1, 100);
} else {
num = getRandomInt(40, 60);
}
$('#out').text(num);
});
function getRandomInt(min, max) {
return Math.floor(Math.random() * (max - min + 1)) + min;
}
});
<script src="https://ajax.googleapis.com/ajax/libs/jquery/1.11.1/jquery.min.js"></script>
<button>Generate</button>
<div id="out"></div>
fiddle: http://jsfiddle.net/kbv39s9w/

I needed to solve this problem a few years ago and my solution was easier than any of the other answers.
I generated 3 randoms between the bounds and averaged them. This pulls the result towards the centre but leaves it completely possible to reach the extremities.

It looks stupid but you can use rand twice:
var choice = Math.random() * 3;
var result;
if (choice < 2){
result = Math.random() * 20 + 40; //you have 2/3 chance to go there
}
else {
result = Math.random() * 100 + 1;
}

Sure it is possible. Make a random 1-100. If the number is <30 then generate number in range 1-100 if not generate in range 40-60.

There is a lot of different ways to generate such random numbers. One way to do it is to compute the sum of multiple uniformly random numbers. How many random numbers you sum and what their range is will determine how the final distribution will look.
The more numbers you sum up, the more it will be biased towards the center. Using the sum of 1 random number was already proposed in your question, but as you notice is not biased towards the center of the range. Other answers have propose using the sum of 2 random numbers or the sum of 3 random numbers.
You can get even more bias towards the center of the range by taking the sum of more random numbers. At the extreme you could take the sum of 99 random numbers which each were either 0 or 1. That would be a binomial distribution. (Binomial distributions can in some sense be seen as the discrete version of normal distributions). This can still in theory cover the full range, but it has so much bias towards the center that you should never expect to see it reach the endpoints.
This approach means you can tweak just how much bias you want.

What about using something like this:
var loops = 10;
var tries = 10;
var div = $("#results").html(random());
function random() {
var values = "";
for(var i=0; i < loops; i++) {
var numTries = tries;
do {
var num = Math.floor((Math.random() * 100) + 1);
numTries--;
}
while((num < 40 || num >60) && numTries > 1)
values += num + "<br/>";
}
return values;
}
<script src="https://ajax.googleapis.com/ajax/libs/jquery/1.11.1/jquery.min.js"></script>
<div id="results"></div>
The way I've coded it allows you to set a couple of variables:
loops = number of results
tries = number of times the function will try to get a number between 40-60 before it stops running through the while loop
Added bonus: It uses do while!!! Awesomeness at its best

You can write a function that maps random values between [0, 1) to [1, 100] according to weight. Consider this example:
Here, the value 0.95 maps to value between [61, 100].
In fact we have .05 / .1 = 0.5, which, when mapped to [61, 100], yields 81.
Here is the function:
/*
* Function that returns a function that maps random number to value according to map of probability
*/
function createDistributionFunction(data) {
// cache data + some pre-calculations
var cache = [];
var i;
for (i = 0; i < data.length; i++) {
cache[i] = {};
cache[i].valueMin = data[i].values[0];
cache[i].valueMax = data[i].values[1];
cache[i].rangeMin = i === 0 ? 0 : cache[i - 1].rangeMax;
cache[i].rangeMax = cache[i].rangeMin + data[i].weight;
}
return function(random) {
var value;
for (i = 0; i < cache.length; i++) {
// this maps random number to the bracket and the value inside that bracket
if (cache[i].rangeMin <= random && random < cache[i].rangeMax) {
value = (random - cache[i].rangeMin) / (cache[i].rangeMax - cache[i].rangeMin);
value *= cache[i].valueMax - cache[i].valueMin + 1;
value += cache[i].valueMin;
return Math.floor(value);
}
}
};
}
/*
* Example usage
*/
var distributionFunction = createDistributionFunction([
{ weight: 0.1, values: [1, 40] },
{ weight: 0.8, values: [41, 60] },
{ weight: 0.1, values: [61, 100] }
]);
/*
* Test the example and draw results using Google charts API
*/
function testAndDrawResult() {
var counts = [];
var i;
var value;
// run the function in a loop and count the number of occurrences of each value
for (i = 0; i < 10000; i++) {
value = distributionFunction(Math.random());
counts[value] = (counts[value] || 0) + 1;
}
// convert results to datatable and display
var data = new google.visualization.DataTable();
data.addColumn("number", "Value");
data.addColumn("number", "Count");
for (value = 0; value < counts.length; value++) {
if (counts[value] !== undefined) {
data.addRow([value, counts[value]]);
}
}
var chart = new google.visualization.ColumnChart(document.getElementById("chart"));
chart.draw(data);
}
google.load("visualization", "1", { packages: ["corechart"] });
google.setOnLoadCallback(testAndDrawResult);
<script src="https://www.google.com/jsapi"></script>
<div id="chart"></div>

Here's a weighted solution at 3/4 40-60 and 1/4 outside that range.
function weighted() {
var w = 4;
// number 1 to w
var r = Math.floor(Math.random() * w) + 1;
if (r === 1) { // 1/w goes to outside 40-60
var n = Math.floor(Math.random() * 80) + 1;
if (n >= 40 && n <= 60) n += 40;
return n
}
// w-1/w goes to 40-60 range.
return Math.floor(Math.random() * 21) + 40;
}
function test() {
var counts = [];
for (var i = 0; i < 2000; i++) {
var n = weighted();
if (!counts[n]) counts[n] = 0;
counts[n] ++;
}
var output = document.getElementById('output');
var o = "";
for (var i = 1; i <= 100; i++) {
o += i + " - " + (counts[i] | 0) + "\n";
}
output.innerHTML = o;
}
test();
<pre id="output"></pre>

Ok, so I decided to add another answer because I felt like my last answer, as well as most answers here, use some sort of half-statistical way of obtaining a bell-curve type result return. The code I provide below works the same way as when you roll a dice. Therefore, it is hardest to get 1 or 99, but easiest to get 50.
var loops = 10; //Number of numbers generated
var min = 1,
max = 50;
var div = $("#results").html(random());
function random() {
var values = "";
for (var i = 0; i < loops; i++) {
var one = generate();
var two = generate();
var ans = one + two - 1;
var num = values += ans + "<br/>";
}
return values;
}
function generate() {
return Math.floor((Math.random() * (max - min + 1)) + min);
}
<script src="https://ajax.googleapis.com/ajax/libs/jquery/1.11.1/jquery.min.js"></script>
<div id="results"></div>

I'd recommend using the beta distribution to generate a number between 0-1, then scale it up. It's quite flexible and can create many different shapes of distributions.
Here's a quick and dirty sampler:
rbeta = function(alpha, beta) {
var a = 0
for(var i = 0; i < alpha; i++)
a -= Math.log(Math.random())
var b = 0
for(var i = 0; i < beta; i++)
b -= Math.log(Math.random())
return Math.ceil(100 * a / (a+b))
}

var randNum;
// generate random number from 1-5
var freq = Math.floor(Math.random() * (6 - 1) + 1);
// focus on 40-60 if the number is odd (1,3, or 5)
// this should happen %60 of the time
if (freq % 2){
randNum = Math.floor(Math.random() * (60 - 40) + 40);
}
else {
randNum = Math.floor(Math.random() * (100 - 1) + 1);
}

The best solution targeting this very problem is the one proposed by BlueRaja - Danny Pflughoeft but I think a somewhat faster and more general solution is also worth mentioning.
When I have to generate random numbers (strings, coordinate pairs, etc.) satisfying the two requirements of
The result set is quite small. (not larger than 16K numbers)
The result set is discreet. (like integer numbers only)
I usually start by creating an array of numbers (strings, coordinate pairs, etc.) fulfilling the requirement (In your case: an array of numbers containing the more probable ones multiple times.), then choose a random item of that array. This way, you only have to call the expensive random function once per item.

Distribution
5% for [ 0,39]
90% for [40,59]
5% for [60,99]
Solution
var f = Math.random();
if (f < 0.05) return random(0,39);
else if (f < 0.95) return random(40,59);
else return random(60,99);
Generic Solution
random_choose([series(0,39),series(40,59),series(60,99)],[0.05,0.90,0.05]);
function random_choose (collections,probabilities)
{
var acc = 0.00;
var r1 = Math.random();
var r2 = Math.random();
for (var i = 0; i < probabilities.length; i++)
{
acc += probabilities[i];
if (r1 < acc)
return collections[i][Math.floor(r2*collections[i].length)];
}
return (-1);
}
function series(min,max)
{
var i = min; var s = [];
while (s[s.length-1] < max) s[s.length]=i++;
return s;
}

You can use a helper random number to whether generate random numbers in 40-60 or 1-100:
// 90% of random numbers should be between 40 to 60.
var weight_percentage = 90;
var focuse_on_center = ( (Math.random() * 100) < weight_percentage );
if(focuse_on_center)
{
// generate a random number within the 40-60 range.
alert (40 + Math.random() * 20 + 1);
}
else
{
// generate a random number within the 1-100 range.
alert (Math.random() * 100 + 1);
}

If you can use the gaussian function, use it. This function returns normal number with average 0 and sigma 1.
95% of this number are within average +/- 2*sigma. Your average = 50, and sigma = 5 so
randomNumber = 50 + 5*gaussian()

The best way to do that is generating a random number that is distributed equally in a certain set of numbers, and then apply a projection function to the set between 0 and a 100 where the projection is more likely to hit the numbers you want.
Typically the mathematical way of achieving this is plotting a probability function of the numbers you want. We could use the bell curve, but let's for the sake of easier calculation just work with a flipped parabola.
Let's make a parabola such that its roots are at 0 and 100 without skewing it. We get the following equation:
f(x) = -(x-0)(x-100) = -x * (x-100) = -x^2 + 100x
Now, all the area under the curve between 0 and 100 is representative of our first set where we want the numbers generated. There, the generation is completely random. So, all we need to do is find the bounds of our first set.
The lower bound is, of course, 0. The upper bound is the integral of our function at 100, which is
F(x) = -x^3/3 + 50x^2
F(100) = 500,000/3 = 166,666.66666 (let's just use 166,666, because rounding up would make the target out of bounds)
So we know that we need to generate a number somewhere between 0 and 166,666. Then, we simply need to take that number and project it to our second set, which is between 0 and 100.
We know that the random number we generated is some integral of our parabola with an input x between 0 and 100. That means that we simply have to assume that the random number is the result of F(x), and solve for x.
In this case, F(x) is a cubic equation, and in the form F(x) = ax^3 + bx^2 + cx + d = 0, the following statements are true:
a = -1/3
b = 50
c = 0
d = -1 * (your random number)
Solving this for x yields you the actual random number your are looking for, which is guaranteed to be in the [0, 100] range and a much higher likelihood to be close to the center than the edges.

This answer is really good. But I would like to post implementation instructions (I'm not into JavaScript, so I hope you will understand) for different situation.
Assume you have ranges and weights for every range:
ranges - [1, 20], [21, 40], [41, 60], [61, 100]
weights - {1, 2, 100, 5}
Initial Static Information, could be cached:
Sum of all weights (108 in sample)
Range selection boundaries. It basically this formula: Boundary[n] = Boundary[n - 1] + weigh[n - 1] and Boundary[0] = 0. Sample has Boundary = {0, 1, 3, 103, 108}
Number generation:
Generate random number N from range [0, Sum of all weights).
for (i = 0; i < size(Boundary) && N > Boundary[i + 1]; ++i)
Take ith range and generate random number in that range.
Additional note for performance optimizations. Ranges don't have to be ordered neither ascending nor descending order, so for faster range look-up range that has highest weight should go first and one with lowest weight should go last.

javascript - generate a new random number

I have a variable that has a number between 1-3.
I need to randomly generate a new number between 1-3 but it must not be the same as the last one.
It happens in a loop hundreds of times.
What is the most efficient way of doing this?

May the powers of modular arithmetic help you!!
This function does what you want using the modulo operator:
/**
* generate(1) will produce 2 or 3 with probablity .5
* generate(2) will produce 1 or 3 with probablity .5
* ... you get the idea.
*/
function generate(nb) {
rnd = Math.round(Math.random())
return 1 + (nb + rnd) % 3
}
if you want to avoid a function call, you can inline the code.

Here is a jsFiddle that solves your problem : http://jsfiddle.net/AsMWG/
I've created an array containing 1,2,3 and first I select any number and swap it with the last element. Then I only pick elements from position 0 and 1, and swap them with last element.

var x = 1; // or 2 or 3
// this generates a new x out of [1,2,3] which is != x
x = (Math.floor(2*Math.random())+x) % 3 + 1;

You can randomly generate numbers with the random number generator built in to javascript. You need to use Math.random().
If you're push()-ing into an array, you can always check if the previously inserted one is the same number, thus you regenerate the number. Here is an example:
var randomArr = [];
var count = 100;
var max = 3;
var min = 1;
while (randomArr.length < count) {
var r = Math.floor(Math.random() * (max - min) + min);
if (randomArr.length == 0) {
// start condition
randomArr.push(r);
} else if (randomArr[randomArr.length-1] !== r) {
// if the previous value is not the same
// then push that value into the array
randomArr.push(r);
}
}

As Widor commented generating such a number is equivalent to generating a number with probability 0.5. So you can try something like this (not tested):
var x; /* your starting number: 1,2 or 3 */
var y = Math.round(Math.random()); /* generates 0 or 1 */
var i = 0;
var res = i+1;
while (i < y) {
res = i+1;
i++;
if (i+1 == x) i++;
}

The code is tested and it does for what you are after.
var RandomNumber = {
lastSelected: 0,
generate: function() {
var random = Math.floor(Math.random()*3)+1;
if(random == this.lastSelected) {
generateNumber();
}
else {
this.lastSelected = random;
return random;
}
}
}
RandomNumber.generate();

Generating unique random numbers (integers) between 0 and 'x'

I need to generate a set of unique (no duplicate) integers, and between 0 and a given number.
That is:
var limit = 10;
var amount = 3;
How can I use Javascript to generate 3 unique numbers between 1 and 10?

Use the basic Math methods:
Math.random() returns a random number between 0 and 1 (including 0, excluding 1).
Multiply this number by the highest desired number (e.g. 10)
Round this number downward to its nearest integer
Math.floor(Math.random()*10) + 1
Example:
//Example, including customisable intervals [lower_bound, upper_bound)
var limit = 10,
amount = 3,
lower_bound = 1,
upper_bound = 10,
unique_random_numbers = [];
if (amount > limit) limit = amount; //Infinite loop if you want more unique
//Natural numbers than exist in a
// given range
while (unique_random_numbers.length < limit) {
var random_number = Math.floor(Math.random()*(upper_bound - lower_bound) + lower_bound);
if (unique_random_numbers.indexOf(random_number) == -1) {
// Yay! new random number
unique_random_numbers.push( random_number );
}
}
// unique_random_numbers is an array containing 3 unique numbers in the given range

Math.floor(Math.random() * (limit+1))
Math.random() generates a floating point number between 0 and 1, Math.floor() rounds it down to an integer.
By multiplying it by a number, you effectively make the range 0..number-1. If you wish to generate it in range from num1 to num2, do:
Math.floor(Math.random() * (num2-num1 + 1) + num1)
To generate more numbers, just use a for loop and put results into an array or write them into the document directly.

function generateRange(pCount, pMin, pMax) {
min = pMin < pMax ? pMin : pMax;
max = pMax > pMin ? pMax : pMin;
var resultArr = [], randNumber;
while ( pCount > 0) {
randNumber = Math.round(min + Math.random() * (max - min));
if (resultArr.indexOf(randNumber) == -1) {
resultArr.push(randNumber);
pCount--;
}
}
return resultArr;
}
Depending on range needed the method of returning the integer can be changed to: ceil (a,b], round [a,b], floor [a,b), for (a,b) is matter of adding 1 to min with floor.

Math.floor(Math.random()*limit)+1

for(i = 0;i <amount; i++)
{
var randomnumber=Math.floor(Math.random()*limit)+1
document.write(randomnumber)
}

Here’s another algorithm for ensuring the numbers are unique:
generate an array of all the numbers from 0 to x
shuffle the array so the elements are in random order
pick the first n
Compared to the method of generating random numbers until you get a unique one, this method uses more memory, but it has a more stable running time – the results are guaranteed to be found in finite time. This method works better if the upper limit is relatively low or if the amount to take is relatively high.
My answer uses the Lodash library for simplicity, but you could also implement the algorithm described above without that library.
// assuming _ is the Lodash library
// generates `amount` numbers from 0 to `upperLimit` inclusive
function uniqueRandomInts(upperLimit, amount) {
var possibleNumbers = _.range(upperLimit + 1);
var shuffled = _.shuffle(possibleNumbers);
return shuffled.slice(0, amount);
}

Something like this
var limit = 10;
var amount = 3;
var nums = new Array();
for(int i = 0; i < amount; i++)
{
var add = true;
var n = Math.round(Math.random()*limit + 1;
for(int j = 0; j < limit.length; j++)
{
if(nums[j] == n)
{
add = false;
}
}
if(add)
{
nums.push(n)
}
else
{
i--;
}
}

var randomNums = function(amount, limit) {
var result = [],
memo = {};
while(result.length < amount) {
var num = Math.floor((Math.random() * limit) + 1);
if(!memo[num]) { memo[num] = num; result.push(num); };
}
return result; }
This seems to work, and its constant lookup for duplicates.

These answers either don't give unique values, or are so long (one even adding an external library to do such a simple task).
1. generate a random number.
2. if we have this random already then goto 1, else keep it.
3. if we don't have desired quantity of randoms, then goto 1.
function uniqueRandoms(qty, min, max){
var rnd, arr=[];
do { do { rnd=Math.floor(Math.random()*max)+min }
while(arr.includes(rnd))
arr.push(rnd);
} while(arr.length<qty)
return arr;
}
//generate 5 unique numbers between 1 and 10
console.log( uniqueRandoms(5, 1, 10) );
...and a compressed version of the same function:
function uniqueRandoms(qty,min,max){var a=[];do{do{r=Math.floor(Math.random()*max)+min}while(a.includes(r));a.push(r)}while(a.length<qty);return a}

/**
* Generates an array with numbers between
* min and max randomly positioned.
*/
function genArr(min, max, numOfSwaps){
var size = (max-min) + 1;
numOfSwaps = numOfSwaps || size;
var arr = Array.apply(null, Array(size));
for(var i = 0, j = min; i < size & j <= max; i++, j++) {
arr[i] = j;
}
for(var i = 0; i < numOfSwaps; i++) {
var idx1 = Math.round(Math.random() * (size - 1));
var idx2 = Math.round(Math.random() * (size - 1));
var temp = arr[idx1];
arr[idx1] = arr[idx2];
arr[idx2] = temp;
}
return arr;
}
/* generating the array and using it to get 3 uniques numbers */
var arr = genArr(1, 10);
for(var i = 0; i < 3; i++) {
console.log(arr.pop());
}

I think, this is the most human approach (with using break from while loop), I explained it's mechanism in comments.
function generateRandomUniqueNumbersArray (limit) {
//we need to store these numbers somewhere
const array = new Array();
//how many times we added a valid number (for if statement later)
let counter = 0;
//we will be generating random numbers until we are satisfied
while (true) {
//create that number
const newRandomNumber = Math.floor(Math.random() * limit);
//if we do not have this number in our array, we will add it
if (!array.includes(newRandomNumber)) {
array.push(newRandomNumber);
counter++;
}
//if we have enought of numbers, we do not need to generate them anymore
if (counter >= limit) {
break;
}
}
//now hand over this stuff
return array;
}
You can of course add different limit (your amount) to the last 'if' statement, if you need less numbers, but be sure, that it is less or equal to the limit of numbers itself - otherwise it will be infinite loop.

Just as another possible solution based on ES6 Set ("arr. that can contain unique values only").
Examples of usage:
// Get 4 unique rnd. numbers: from 0 until 4 (inclusive):
getUniqueNumbersInRange(4, 0, 5) //-> [5, 0, 4, 1];
// Get 2 unique rnd. numbers: from -1 until 2 (inclusive):
getUniqueNumbersInRange(2, -1, 2) //-> [1, -1];
// Get 0 unique rnd. numbers (empty result): from -1 until 2 (inclusive):
getUniqueNumbersInRange(0, -1, 2) //-> [];
// Get 7 unique rnd. numbers: from 1 until 7 (inclusive):
getUniqueNumbersInRange(7, 1, 7) //-> [ 3, 1, 6, 2, 7, 5, 4];
The implementation:
function getUniqueNumbersInRange(uniqueNumbersCount, fromInclusive, untilInclusive) {
// 0/3. Check inputs.
if (0 > uniqueNumbersCount) throw new Error('The number of unique numbers cannot be negative.');
if (fromInclusive > untilInclusive) throw new Error('"From" bound "' + fromInclusive
+ '" cannot be greater than "until" bound "' + untilInclusive + '".');
const rangeLength = untilInclusive - fromInclusive + 1;
if (uniqueNumbersCount > rangeLength) throw new Error('The length of the range is ' + rangeLength + '=['
+ fromInclusive + '…' + untilInclusive + '] that is smaller than '
+ uniqueNumbersCount + ' (specified count of result numbers).');
if (uniqueNumbersCount === 0) return [];
// 1/3. Create a new "Set" – object that stores unique values of any type, whether primitive values or object references.
// MDN - https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Set
// Support: Google Chrome 38+(2014.10), Firefox 13+, IE 11+
const uniqueDigits = new Set();
// 2/3. Fill with random numbers.
while (uniqueNumbersCount > uniqueDigits.size) {
// Generate and add an random integer in specified range.
const nextRngNmb = Math.floor(Math.random() * rangeLength) + fromInclusive;
uniqueDigits.add(nextRngNmb);
}
// 3/3. Convert "Set" with unique numbers into an array with "Array.from()".
// MDN – https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Array/from
// Support: Google Chrome 45+ (2015.09+), Firefox 32+, not IE
const resArray = Array.from(uniqueDigits);
return resArray;
}
The benefits of the current implementation:
Have a basic check of input arguments – you will not get an unexpected output when the range is too small, etc.
Support the negative range (not only from 0), e. g. randoms from -1000 to 500, etc.
Expected behavior: the current most popular answer will extend the range (upper bound) on its own if input bounds are too small. An example: get 10000 unique numbers with a specified range from 0 until 10 need to throw an error due to too small range (10-0+1=11 possible unique numbers only). But the current top answer will hiddenly extend the range until 10000.

I wrote this C# code a few years back, derived from a Wikipedia-documented algorithm, which I forget now (feel free to comment...). Uniqueness is guaranteed for the lifetime of the HashSet. Obviously, if you will be using a database, you could store the generated numbers there. Randomness was ok for my needs, but probably can be improved using a different RNG. Note: count must be <= max - min (duh!) and you can easily modify to generate ulongs.
private static readonly Random RndGen = new Random();
public static IEnumerable<int> UniqueRandomIntegers(int count, int min, int max)
{
var rv = new HashSet<int>();
for (var i = max - min - count + 1; i <= max - min; i++)
{
var r = (int)(RndGen.NextDouble() * i);
var v = rv.Contains(r) ? i : r;
rv.Add(v);
yield return v;
}
}

Randomized Array, Sliced
Similar to #rory-okane's answer, but without lodash.
Both Time Complexity and Space Complexity = O(n) where n=limit
Has a consistent runtime
Supports a positive or negative range of numbers
Theoretically, this should support a range from 0 to ±2^32 - 1
This limit is due to Javascript arrays only supporting 2^32 - 1 indexes as per the ECMAScript specification
I stopped testing it at 10^8 because my browser got weird around here and strangely only negative numbers to -10^7 - I got an Uncaught RangeError: Invalid array length error (shrug)
Bonus feature: Generate a randomized array of n length 0 to limit if you pass only one argument
let uniqueRandomNumbers = (limit, amount = limit) => {
let array = Array(Math.abs(limit));
for (let i = 0; i < array.length; i++) array[i] = i * Math.sign(limit);
let currentIndex = array.length;
let randomIndex;
while(currentIndex > 0) {
randomIndex = Math.floor(Math.random() * currentIndex--);
[array[currentIndex], array[randomIndex]] = [array[randomIndex], array[currentIndex]];
}
return array.slice(0, Math.abs(amount));
}
console.log(uniqueRandomNumbers(10, 3));
console.log(uniqueRandomNumbers(-10, 3));
//bonus feature:
console.log(uniqueRandomNumbers(10));
Credit:
I personally got here because I was trying to generate random arrays of n length. Other SO questions that helped me arrive at this answer for my own use case are below. Thank you everyone for your contributions, you made my life better today.
Most efficient way to create a zero filled JavaScript array?
How to randomize (shuffle) a JavaScript array?
Also the answer from #ashleedawg is where I started, but when I discovered the infinite loop issues I ended up at the sliced randomized array approach.

const getRandomNo = (min, max) => {
min = Math.ceil(min);
max = Math.floor(max);
return Math.floor(Math.random() * (max - min + 1)) + min;
}
This function returns a random integer between the specified values. The value is no lower than min (or the next integer greater than min if min isn't an integer) and is less than (but not equal to) max.
Example
console.log(`Random no between 0 and 10 ${getRandomNo(0,10)}`)

Here's a simple, one-line solution:
var limit = 10;
var amount = 3;
randoSequence(1, limit).slice(0, amount);
It uses randojs.com to generate a randomly shuffled array of integers from 1 through 10 and then cuts off everything after the third integer. If you want to use this answer, toss this within the head tag of your HTML document:
<script src="https://randojs.com/1.0.0.js"></script>