If I have relative probabilities of events A, B, C occurring.

i.e P(A) = 0.45, P(B) = 0.35, P(C) = 0.20,

How do I do represent this using a random number generator between 0 and 1?

i.e. R = rand(0,1)

if (R < 0.45)
   event A

else if(R < 0.35)
  event B

else if(R < 0.20)
  event C

The above works for two events A,B but I think the above is wrong for three or more since there is overlapping.

This is obviously a very simple question and the answer should be immediately evident, but I'm just too stupid to see it.