THINK LIKE A DATA SCIENTIST:
Probability is hard because counting is hard.
A thread. π
Probability is hard because counting is hard.
A thread. π
For a lot of people, mathematics is true in the same way that "Kermit The Frog and Miss Piggy are a couple" is true. It's true in an imaginary world where we have agreed upon rules. If that's how you think about math then it's pretty obvious that "2+2=4".
To me, "2+2=4" means that "2 things + 2 things will always be 4 things no matter what the things are". Turns out this is not technically true. You can create all kinds of mathematical systems and physical situations where 2 things + 2 things is not 4 things.
For instance, imagine you're sitting at a table. I give you 2 beakers with 2 cups of a clear fluid in each one. You add them both to a big measuring cup and find that it's actually less than 4 cups.
Turns out one beaker had ethanol in it and the other had water. When you mix those two fluids together, it turns out that they interact in an unexpected way.
You might argue that "2 cups of clear fluid plus 2 cups of clear fluid is always four cups of clear fluid" is false but "2 cups of water plus 2 cups of water is always 4 cups of water" isn't. This might actually be the case but I would argue that we don't know even this.
What if some new physical effect kicks in that we don't know about yet? Then we'd be back in the same boat as before of trying to come up with a way of redefining the situation so that "2+2=4" makes sense. We have to be physicists to figure out if we can trust our arithmetic.
The problem might be the mixing. Perhaps we should have a rule that as long as we don't let the volumes of liquid that we're adding up physically interact then we'll have 4 cups of fluid just not all in the same container.
So perhaps with the right rules about how to set things up, we can always make them line up. I notice we humans do that a lot with arithmetic. We make excuses for it. If things don't add up in the way we were expecting, we fiddle around with our definitions until it works out.
Imagine you're hiring four people to help cater a wedding. Two very trusted recruiters each send you a list of two excellent candidates so you move ahead with your plans, only to find out just before the wedding that "Elizabeth Jones" on list one is also "Beth Jones" on list two!
In this case, the "interaction" wasn't physical. It was conceptual. We intended to count how many people we had but instead we were just counting names. Because the same name can refer to multiple people & vice versa, we ended up in an unexpected situation.
When you learn "2 apples + 2 apples = 4 apples" in grade school, they don't bring up these kinds of problems. A pair of apples can't be, for instance, a projection into our universe of a single apple that exists in a higher dimensional space. I know I sound crazy right now but... 
When I give examples like these, the main objection folks make is that I'm using "+" and "=" in a different way than they're meant to be used. When somebody makes this kind of objection, it's clear to me that they look at math in a Kermit-and-Miss-Piggy way.
For them, "2+2=4" is just statement about rules in an imaginary world. It doesn't matter to them that it doesn't apply to some stuff in our world. "2+2=4" is always true because it lives in a perfect realm of abstract forms where it is untarnished by our fallen world.
To a Kermit-and-Miss-Piggy type person, we've all agreed that the symbols "2+2=4" must always refer to things in the context of that ideal world. Therefore, I'm breaking the social contract when I use "2+2=4" in the context of our world.
"Breaking the social contract" is another way of saying that I'm being an asshole. In case that's how you're feeling right now, let me try and defend myself a little.
One of the reasons I got obsessed with thinking about things like this is I'm a data analyst. A big part of analyzing data is just counting stuff. If you can't answer basic questions like how many things you have, it's really hard to figure out the probabilities of things.
Turns out how many "things" you have can be really dependent on your definition of a "thing" as I hope I've convinced you by now. Lots of data scientists work with huge datasets. Think about a company with billions of customers. It can take days or even weeks to count things.
This sort of thing makes you want to think really carefully about what you're counting because messing up is expensive. It's in this context that I learned an important lesson about definitions. How much of something I had depended a lot on my definitions.
This really important lesson transfers to probability because probability comes from counting how often stuff happens. One of the reasons people don't understand probability is they don't understand how incredibly sensitive both counting and probability are to their definitions.
I won't go into tricky examples with probability right now because this thread is long enough but I wanted to explain the basic challenges using something simple like "2+2=4" which is easier to understand. Hope you found this thread helpful. π
If you enjoyed the thread, please like and retweet so others can enjoy it as well. I talk about this stuff all the time so give me a follow for more of the same.π
P.S. My apologies for the anti-intellectual tone of some of the responses. Don't worry. I'm used to it.π
P.P.S. The thread was retweeted by some large troll accounts. It's mostly bad faith feedback by people that were looking to lash out before they even saw the thread. It's best to ignore it. π
https://twitter.com/bwahacker/status/1426340793594355712?s=20
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