Random number generator with higher probabilities of giving low values?

Question

How can I generate a pseudo-random number (preferably in Lua), where the generator has a higher probability of giving small numbers?

In my case, I want to give a random score in a game, where it's common to obtain the lower scores, but higher ones appear rarely. I've seen weighted random number generators that use a table, but it doesn't fit my plan. I just want to specify the minimum (0) the maximum (variable) and ensure most numbers stay low.

I am sure this is possible with a simple mathematical operation, but I cannot remember which one it was. Like to filter the regular output of math.random, no need for a really random generator.

Answer 1

Try math.floor(minscore+(maxscore-minscore)*math.random()^2). Adjust the power to suit your desired distribution.

Answer 2

I found Ihf's answer very useful, and I create a C# method for it:

    private int GetRandomNumber(int max, int min, double probabilityPower = 2)
    {
        var randomizer = new Random();
        var randomDouble = randomizer.NextDouble();

        var result = Math.Floor(min + (max + 1 - min) * (Math.Pow(randomDouble, probabilityPower)));
        return (int) result;
    }

If probabilityPower is above 1, lower values will be more common than higher values. If it's between 0 to 1, higher values will be more common than lower values. If it's 1, the results will be in a general randomness.

Examples (all with 1 million iterations, min = 1, max = 20 ):

---

probabilityPower = 1.5

1: 135534 (13.5534%)
2: 76829 (7.6829%)
3: 68999 (6.8999%)
4: 60909 (6.0909%)
5: 54595 (5.4595%)
6: 53555 (5.3555%)
7: 48529 (4.8529%)
8: 44688 (4.4688%)
9: 43969 (4.3969%)
10: 44314 (4.4314%)
11: 40123 (4.0123%)
12: 39920 (3.992%)
13: 40466 (4.0466%)
14: 35821 (3.5821%)
15: 37862 (3.7862%)
16: 35222 (3.5222%)
17: 35902 (3.5902%)
18: 35202 (3.5202%)
19: 33961 (3.3961%)
20: 33600 (3.36%)

---

probabilityPower = 4

1: 471570 (47.157%)
2: 90114 (9.0114%)
3: 60333 (6.0333%)
4: 46574 (4.6574%)
5: 38905 (3.8905%)
6: 32379 (3.2379%)
7: 28309 (2.8309%)
8: 27906 (2.7906%)
9: 22389 (2.2389%)
10: 21524 (2.1524%)
11: 19444 (1.9444%)
12: 19688 (1.9688%)
13: 18398 (1.8398%)
14: 16870 (1.687%)
15: 15517 (1.5517%)
16: 15871 (1.5871%)
17: 14550 (1.455%)
18: 14635 (1.4635%)
19: 13399 (1.3399%)
20: 11625 (1.1625%)

---

probabilityPower = 1

1: 51534 (5.1534%)
2: 49239 (4.9239%)
3: 50955 (5.0955%)
4: 47992 (4.7992%)
5: 48971 (4.8971%)
6: 50456 (5.0456%)
7: 49282 (4.9282%)
8: 51344 (5.1344%)
9: 50841 (5.0841%)
10: 48548 (4.8548%)
11: 49294 (4.9294%)
12: 51795 (5.1795%)
13: 50583 (5.0583%)
14: 51020 (5.102%)
15: 51060 (5.106%)
16: 48632 (4.8632%)
17: 48568 (4.8568%)
18: 50039 (5.0039%)
19: 49863 (4.9863%)
20: 49984 (4.9984%)

---

probabilityPower = 0.5

1: 3899 (0.3899%)
2: 5818 (0.5818%)
3: 12808 (1.2808%)
4: 17880 (1.788%)
5: 23109 (2.3109%)
6: 26469 (2.6469%)
7: 33435 (3.3435%)
8: 35243 (3.5243%)
9: 42276 (4.2276%)
10: 47235 (4.7235%)
11: 52907 (5.2907%)
12: 58107 (5.8107%)
13: 63719 (6.3719%)
14: 66266 (6.6266%)
15: 72708 (7.2708%)
16: 79257 (7.9257%)
17: 81830 (8.183%)
18: 87243 (8.7243%)
19: 90958 (9.0958%)
20: 98833 (9.8833%)

---

probabilityPower = 0.4

1: 917 (0.0917%)
2: 3917 (0.3917%)
3: 3726 (0.3726%)
4: 10679 (1.0679%)
5: 13092 (1.3092%)
6: 17306 (1.7306%)
7: 22838 (2.2838%)
8: 29221 (2.9221%)
9: 35832 (3.5832%)
10: 38422 (3.8422%)
11: 47800 (4.78%)
12: 53431 (5.3431%)
13: 63791 (6.3791%)
14: 69460 (6.946%)
15: 75313 (7.5313%)
16: 86536 (8.6536%)
17: 95082 (9.5082%)
18: 103440 (10.344%)
19: 110203 (11.0203%)
20: 118994 (11.8994%)