# Peer Pressure - Drinking

Comes from a class I took senior year of HS, and I didn't find it online so hopefully this isn't a repeat.

A renowned psychologist is touring High Schools all over the country with her latest finding.

Theory: There is a really popular student in each class in the entire school, such that if that student drinks, then everyone in that class drinks.

Is this just due to a small sample size, or is this always true, sometimes true, or never true? And please prove why this is the case.

Also please feel free to suggest tags. I am not too familiar with the tags here

• Is this a lateral-thinking question at all? Are you looking for a silly answer like "everyone in all classes drinks because without water you'll die"? Jul 17 '15 at 18:16
• That student is taking a lot of classes at once, some hermione granger level stuff. Jul 17 '15 at 18:31
• The drinker's paradox. I'm surprised that hasn't come up on this site, yet :D Jul 17 '15 at 18:56

Let the popular student be named "P" for short. The statement:

If P drinks, then everyone in that class drinks

is the same as the statement

Either P doesn't drink, or everyone in that class drinks.

The existence of a person P making the above statement true is now more clear.

If everyone drinks, choose P to be anybody. If someone doesn't drink, choose P to be that person.

So, the Theorem is always true. This is a famous logical "paradox", it even has a Wikipedia page.

Addendum: It depends on the definition of "really popular". If a student $x$ being "really popular" means "if $x$ drinks, then everyone drinks", then the above reasoning is correct. If "really popular" is defined independently of the drinking condition, then the Theory is not true in general. A counterexample is any class with an unpopular student who doesn't drink, while everyone else drinks (believe me, such classes exist).

• What about when the person that doesn't drink isn't a popular student? Jul 17 '15 at 18:22
• What if there are 2 students who don't drink? At any time, at least one of those two will not be P, and the statement will not be true. Jul 17 '15 at 18:26
• @CodeNewbie Then choose one of them arbitrarily. Jul 17 '15 at 18:28
• Reading the statement, it sounds like "popular" is being defined by the rest of the question. In other words, "popular" means "dictating the drinking-ness of the class". Jul 17 '15 at 18:31
• @CodeNewbie Did you see the Wikipedia page he linked to? It works like this: represent the statement as $P \to Q$, where $P$ is "popular student drinks" and $Q$ is everyone drinks. If the popular student doesn't drink, $P$ is obviously false. Since $P$ isn't true, $P \to Q$ is assumed to be true. So in your example, assume you are the popular student. Since you don't drink, it doesn't matter if you will drink or you won't drink. We can assume the statement is true, and like the statement says, since you're not drinking, it will never happen that all of you are drinking. Jul 17 '15 at 18:40