There are two things wrong with the accepted answer:
First, while the unconditional counts are all positive, the conditional counts (count of word given context) can be zero.
Second, the sum of the counts of words given context is not equal to the unigram count! It is equal to the total number of bigrams that begin with that context. These are two completely different numbers.
I am not sure if this is truly a solved problem but I am very sure about those two points.