Mysterious Ratio

Chef has been teaching probability and expectation in his class. He wants his $N$ students (numbered $1$ through $N$) to play a game so that they could understand the concepts clearly. For each valid $i$, the $i$-th student should choose an integer from a range $[L_i, R_i]$ uniformly randomly. Chef wants to know the expected value of the ratio $A/B$, where $A$ is the product of the integers cho