A conditional random number generation problem (please help me!)

I need to know the formula for the random function that return random
numbers in a range of a and b integers [a,b] but that obey on a custom
probability (possibly different!) for each integer number on this [a,b]
range (of course the sum of all integer number probabilities are = 1!).
Finally, what i want is the general function formula that simulate the
random behavior (based on a custom probability value for each integer
number between the [a,b] range. confuse? i hope not! please help me!!!!

what i know so far is that the function formula for generating a "pure"
random number between [a,b] range is:

rand()*(b-a)+a

where rand() return a random number between 0 and 1.

.

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