If $\lambda_1,\dots,\lambda_n$ are distinct positive real numbers, then $$\sum_{i=1}^n \prod_{j\neq i} {\lambda_j\over \lambda_j-\lambda_i}=1.$$ This identity follows from a probability calculation that you can find at the top of page 311 in the 10th edition of Introduction to Probability Models by Sheldon Ross.
Is there a slick or obvious explanation for this identity?
This question is sort of similar to my previous problem; clearly algebra is not my strong suit!