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

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.

• There are many, but the one posted is pretty good, and the word "unexpected" in the title sets the bar high. How "unexpected" should the solution be? For instance, same birthday problem with lin. expectation and indicator variables, exotic enough? Aug 5 at 17:39
• @AntoniParellada Unexpected in the sense, not the very first idea you get if you saw the problem for the first time(might never), but when told the solution one feels stupid for their cumbersome mathematics calculations :-). There are not any restrictions but yeah less known examples are mostly expected. Aug 6 at 0:29

## 1 Answer

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.

• That's a nice one +1, I drew Venn diagram and wrote inequalities, but your solution is far more beautiful. Aug 4 at 2:10