Unexpected use of linearity of expectation with indicator random variable in problems

Question

Can people suggest some problems (probability puzzle type) where the use of linearity of expectation together with indicator random variable is unexpected/hard to see but it makes problems much easier?

I have encountered a lot of questions asking similar types of problems in the various domains so I think combined use of linearity of expectation and indicator random variable deserves its own. This and This are the question I was motivated from, but some of problem in my second mentioned problem are not puzzle related.

Answer 1

In a certain village 80% of the villagers drink tea, 60% drink beer, and 60% drink wine. Nobody consumes all three beverages. What percentage of villagers consume alcohol?

Let $I_T$ be the indicator that a villager consumes tea, and similarly for $I_B$ and $I_W$. Then the number of beverages consumed by a villager is $$N:=I_T + I_B + I_W.$$ By calculation, using linearity of expectation, $E(N)=2$. But by hypothesis, $N\le 2$. Hence each villager consumes exactly two beverages, at least one of which must be alcoholic.