*Disagreement isn't about taking sides. It's about probability.*

In my last job I had a difficult relationship with my advisor. The difficulty was that whenever I brought an idea to her, she would shoot it down. Whatever I thought the right answer was, she disagreed.

At first I thought the issue had to do with our relationship. I took it personally. Why is she always opposed to my ideas? But then I realized that issue didn't reflect our relationship. It was a simple matter of probability.

Usually we think of disagreement and agreement as a fifty-fifty split. If I say, “chocolate milkshakes are better than vanilla milkshakes,” then the probability that you’ll agree with me is 50%. The assumption that agreement and disagreement are split fifty-fifty is an example of what statisticians call the naive definition of probability.

In formal terms, the naive definition of probability is the assumption that if you have an outcome A and an outcome B, then the probability that A will occur is 50%. Sometimes this assumption makes sense, like if you’re flipping a coin. Either outcome, heads or tails, is equally likely. But for more complicated situations, this assumption doesn’t hold. For instance, what’s the probability that there’s life on mars? There are two outcomes, life or no-life, so under the naive definition it’s fifty-fifty.

But the chance that there's life on mars isn't actually 50%. That's not how probability works. The probability that any given planet isn’t going to have life is vastly more likely than a planet having life, regardless of which planet we’re talking about, because we know of millions of planets and none of them appear to harbor life. When we want to know how likely something is, we can’t just look at the number of outcomes. We have to look at how likely each outcome is on its own.

So how likely are agreement and disagreement on their own? If we want to know whether two people agree that chocolate milkshakes are better than vanilla, then it’s reasonable to think that agreement is a fifty-fifty chance. But that’s the simplest possible case. What if, instead, we ask which is your favorite flavor out of out all possible flavors? With an open ended question like this, agreement becomes way less likely than disagreement. One person might say chocolate, but the next might choose peanut butter banana, and the next might choose strawberry. The more complicated the world that you're considering, the more likely disagreement becomes. For almost any topic that comes up in human relationships, agreement and disagreement are not equally likely. Disagreement is the default.

The problem with using the naive model of agreement is that we use it to categorize people into on-our-side and not-on-our-side. If you like the same kind of milkshakes you’re with me, otherwise you’re against me. But if we shift away from naive model of agreement, then it changes the way we think about disagreement. It’s no longer the case that if someone disagrees with you that they’re against you in any meaningful sense. Disagreement isn’t a negative position, just the natural one. The more complicated the subject, the more likely that people disagree about it.

Agreement, then, is rather precious. It’s something to be sought, not just stumbled upon.

When I changed my model of disagreement, then it changed the way I related to my advisor. When she took issue with one of my ideas, it wasn’t necessarily because the idea was a bad one or that she was against my proposals. It was because the problem that we were trying to solve was complicated. Disagreement is the default setting, and agreement is something that you have to work towards. It’s not a statement of taking sides. It’s the probabilistic consequence of living in a complicated world.