Double dixie cup problem doubt I am reading this article: < I have problem with this sentence: "Let $p_i$ be the probability of failure of obtaining m sets up to and including the purchase of the i_th dixie cup. Then the expected number of dixie cups $E_m(n)= \sum_{i=0}^{\inf}p_i$, by a well-known argument ([1] p. 211)." The reference is: "W. Feller, Introduction to Probability Theory, vol.I" But in my version of the book at page 211 contains a different subject. Can someone help me with this "well-know argument"?

If $(X_i)_i$ is a sequence of Bernoulli random variables then the number of successes is $S:=\sum_{i=1}^{\infty}X_i$.

Then by linearity of expectation: $$\mathbb ES=\sum_{i=1}^{\infty}\mathbb EX_i=\sum_{i=1}^{\infty}p_i$$

where $p_i:=P(X_i=1)$.

Does this help (I did not read the article)?