Where 1/3 of the 1 section had been painted, have each section have 3 parts. So each complete section will be 3/3. Where there is 4 section with 3 parts each, we have 12 parts. Thus, you painted 1/12 of the fence.
The binomial distribution requires a constant and known probability for all trials.
For example, if picking marbles from a bag, the marbles must be returned to the bag to achieve a constant probability (with replacement).
The hypergeometric distribution assumes two different classes of objects, and samples are taken out without replacing back in the "pool" (without replacement).
This is especially useful for small populations without replacement, or events having very small probability.