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x + (rand() % y - x + 1) this is a formula to find a random number between x and y but I don't know how it was made I want a logical/mathematical explanation for it and thank you in advance.

François Andrieux
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Ahmed Aboalasaad
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    It's incorrect, it should be `x + (rand() % (y - x + 1))`. – HolyBlackCat Apr 01 '20 at 14:19
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    Because you've left out parentheses, the formula will produce a value between `1` and `y`. The `x` terms cancel out. – Peter Apr 01 '20 at 14:25
  • Even with corrections, you should avoid this algorithm due to [modulo bias](https://stackoverflow.com/questions/10984974/why-do-people-say-there-is-modulo-bias-when-using-a-random-number-generator). – pjs Apr 01 '20 at 15:49

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First, your code should be corrected to x + (rand() % (y - x + 1))

The rand function generates a pseudo random positive integer between 0 and RAND_MAX which is typically implemented by PMMLCG algorithm by Hutchinson. When we assume RAND_MAX is large enough, then the occurrence possibilities of any positive integer are the same. And so, when you apply the result to a mod operation, let's say, %n, then we will have a pseudo random integer in the interval [0, n-1]

In your example, (rand() % (y - x + 1)) generates a pseudo random integer between [0, y-x] and when we add x on it, we get a pseudo random between [x, y]

However, as a supplement, this algorithm has what I would describe as severe problems, the most obvious of which is it relies on the value of RAND_MAX. The standard only guarantee that it's a integer not less than 32767, which is a very small value and so it can contribute to the unevenness of the occurrence possibility of the candidates in the interval.

I suggest you learn to use <random> library in C++11:https://en.cppreference.com/w/cpp/header/random

con ko
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  • @FrançoisAndrieux I had thought to add "typically", but somehow I forgot... Thanks for pointing out. – con ko Apr 01 '20 at 15:20