Does anyone know how to modify that random number generator, so that it will get only one lamp on?
I can't use command blocks.
Probably the easiest way to do what you want is to use the contraption you linked to, and hook it up to a binary decoder. This answer will at least get you started with that, but it seems there are designs that are a bit more compact, but slower. For instance, I found this video which does the same thing as the linked answer. Then, the outputs from the binary decoder just need to be inverted, and you'll have your single active line.
There is an issue though, and that is that the inputs and outputs from the decoder must be separated. The input isn't so much of a problem, but it means it's hard to have the output lamps all in a row. It's doable, but by no means is it easy or compact. Another issue is that the number of outputs must be a power of 2 (1, 2, 4, 8, etc.), or else you'll sometimes have no output.
If you want something more long-termed, just put a chicken in a 2x1 hole with a pressure plate on one floor block. Then connect the pressure plate to your lamp with redstone and you're done.
If you want something usable more often, use a dispenser loaded with stackable and non-stackable items. When the dispenser receives a signal, it drops an item on a weighted pressure plate. If it drops a stackable item, the signal output has power 1; if the item is non-stackable, it outputs 2. This means that you will have two redstone wires in between the lamp/repeater and the pressure plate. If the dispenser is filled right, you get a 4 or 5 out of 9 chance that the lamp turns on. Hope this helped.