If you’re good at solving word problems from elementary math, you’ve probably answered “2” (or even if you weren’t great at word problems, since this is a particularly simple example of such a problem).
If you’re an “out-of-the-box” thinker, you might have said “not sure, but maybe zero” since I didn’t specify where the ice cubes were stored, what the temperature was (did the reference to Thai iced tea mean there’s a high probability that Alice is in relatively hot Thailand?), or how long before Alice takes her afternoon tea.
Why are you asking us silly math questions?
Strictly speaking, the title question isn’t really a math problem. For example, if you had talked to Alice this evening, she might have told you that the weather was so hot that all her ice melted before she could put it in her afternoon tea.
The question only becomes a mathematically solvable problem if we’re able to capture all the relevant information needed in a way that we can represent with math. You may even feel like I was “cheating” when I asked the question, since it sounded like I was asking a question where I was implicitly promising that all the relevant information was there, so all you needed to do was apply your knowledge of math to solve the problem.
Mathematical reasoning is an incredibly powerful tool and has led us to advances in many areas of science. But because it has solved so many problems, it is easy to be persuaded by arguments that use mathematical arguments. After all, you can work through the math and see that all the math is correct.
But what I was trying to point out with my original question is that it’s very easy when constructing a mathematical model to miss elements of the real problem you’re trying to analyze, resulting in a model that is persuasive because of its mathematical correctness, but is completely wrong as a predictive model of the real world.
The more complex the system you’re trying to model with math, the more likely you are to miss important elements that destroy the predictive power of the model. Mathematical models that attempt to predict human behavior are particularly prone to flaws because of the many factors that influence human behavior, making it difficult to capture all these factors in the model. Despite this weakness, there are endless attempts to use math to model human behavior, because there’s a lot of money to be made from correctly predicting future human behavior (stocks, politics, etc).
Ok, sure, but does this result in real world problems?
The real estate bubble and resulting financial crisis of 2007-2008 had their origins in a mathematical model that improperly predicted the “safety” of financial derivatives known as collateralized debt obligations (CDOs for short).
The CDO model was just complicated enough to look reasonable, but still failed to adequately model human behavior. Among other things, it failed to model the fact that the use of this formula itself as a means of valuing CDOs would encourage a loosening in standards for home lending rules that ultimately increased the amount of home owners who could not or would not maintain payments on their home loans when they encountered financial distress.
Ok, so should we just distrust all math models?!?
No, math models are great tools and have lots of uses. But the next time you see someone predicting human behavior based on a math model, I suggest the best place to look for flaws isn’t in the math itself. Look for things the model may miss.
And be very skeptical about the ability of that model to capture all the things that go into a human’s decision making process.