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# patrickwonders

## Patrick

#### Null Intersection Hypothesis?

Here's the situation. You're in an all-day meeting at work. It comes time to order the pizza for lunch. A quick survey of the 20 people present reveals that four of you are vegetarian. Obviously, since 20% of the people are vegetarian, 20% of the pizzas should be meat-free.

Of course, this fails to take into account the fact that some non-vegetarians will have just a slice of the meat-free pizza.

There should be a name for this problem.

The first thing that comes to mind for me is the law of the excluded middle. According to the classical laws of thought, every proposition is either true or not true. There is no middle ground. For this situation, things would have to be framed as: That every person either only eats meat-pizza or only eats non-meat-pizza. That doesn't quite work for me. This suggests the name The Law of the Excluded Eaters.

The next thing that comes to mind is Bayes' Theorem. According to Bayes' Theorem, the probability that someone is vegetarian given they are eating cheese pizza P(V|C) is equal to the (prior) probability that someone is vegetarian P(V) times the probability that someone is eating cheese pizza given they are vegetarian P(C|V) divided by the (prior) probability that someone is eating cheese pizza P(C). The pizza problem is a common fallacy that makes grokking Bayes' Theorem tough for people. The common fallacy is called Berkson's Paradox and is related to the Prosecutor's Fallacy. People inadvertently equate the probability of eating cheese pizza P(C) with the probability that one is vegetarian P(V). This suggests the name The Bayesian Pizza Paradox.

The next thing that comes to mind for me is a simple Venn diagram. The problem assumes that the set of people who eat meat-pizza and the set of people who eat non-meat pizza have zero members in common. The intersection is the Null Set. This suggests the name The Null Intersection Hypothesis.

I like the name, too, because of its association with the Null Hypothesis from statistics. It suggests that every group-pizza order is a sociological experiment where the assumption going in is that meat eaters will eat only meat-pizza and non-meat eaters will eat only non-meat pizza.

The Venn diagram concept also brings up the Inclusion-Exclusion Principle. By that principal, the number of people in the who eat either sausage pizza or cheese pizza |S ∪ C| is equal to the number of people who eat sausage pizza |S| plus the number of people who eat cheese pizza |C| minus the number of people who eat both sausage and cheese pizza |S ∩ C|. It is common for people to forget to subtract that last term. This works when the intersection is empty. This suggests the name The Exclusion-Exclusion Principle.

That same principle here is also related to the Triangle Inequality. By the triangle inequality, the number of people total is less than or equal to the number who eat only meat pizza and the number who eat only non-meat pizza. This name is suggestive in shape. But, I'm not sure the Pizza Slice Inequality really works for me.

Another thing that comes to mind for me is the 80-20 rule. In this case, though, it would be the 80-80 rule: 80% of the people eat 80% of the pizza. It doesn't really work for me though. It doesn't fit well enough.

Another thing that comes to mind is proportional, representative democracy. One person = one vote. This suggests the name Representative Pizzocracy. But, it's not mathy enough for me.

Unless someone has a better suggestion, I'm going with the Null Intersection Hypothesis.

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