I have a bit of a issue with an article came out a while ago on Cracked.com titled, “6 Insane True Statistics That Laugh In The Face of Logic”. While the title is total click-bait, and I recognize that, it still makes me angry. Statistics don’t laugh in the face of logic. The fact that statistics are logical is kind of the whole point of mathematics. Perhaps they are not intuitive and/or not obvious, but they are certainly logical.
The article has several examples of probability & statistics brain teasers that most students will see in an undergraduate Intro to Probability course. This article could have been titled, “6 unintuitive examples your professor will use to stump you on an exam”. I also think it could be called, “A statisticians take on Hunger Games”. Let’s look at the “Insane True Statistics” they cover through the lens of someone who has taken a few math classes.
1. Probability Dictates that “miracles” are routine.
The odds of me winning the lottery twice in a row are tiny (especially since I don’t buy tickets). But the odds of someone winning the lottery twice in a row are pretty good. The odds of Katniss’s little sister being pulled into the Hunger Games were small (she only had one token in the bowl after all). But the odds that someone from district 12 was going to the Hunger Games are 100%. Sample Size is important. With a large enough sample, lots of things with small probability will happen.
Did you ever hear the idea that if you put enough monkeys in a room with enough typewriters that eventually they will produce Hamlet? Theoretically, if one ignores all the tasks of training/feeding monkeys, would this be possible?
2. The odds of two people sharing a birthday in a small group is almost a certainty.
This is a feature that happens in groups due to the fact that there are only 365 days a year. It has to do with Independent vs Dependent variables. It has a wikipedia page. I have nothing else to add.
3. The probability that a man’s sibling is also a male is one in three (not 50-50)
The solution to this is Conditional Probability. If you know the definition of conditionally probability, you too can compute this.
4. You can rig a game of coin flips just by going second
Here too, the solution is to use Conditional Probability. As it turns out, conditional probability is a great source of non-intuitive puzzles. But thankfully, mathematics has rules for this kind of thing and statisticians figured this stuff out a long time ago. Back to de Moivre in in the 18th Century or perhaps even Pascal and Fermat in in the 17th Century. So I’m not sure if this should really “laugh in the face of logic”. Perhaps it merely laughs in the face of people who haven’t seen conditional probability. But that seems like a far cry from fundamentally deposing logic! While these problems are not obvious, I believe that they are totally solvable with persistence.
5. Pi can be calculated by randomly dropping a bunch of paper clips.
Okay, I admit, this one is awesome and seemingly magical even when you understand what is happening. This is true and a very cool experiment called Buffon’s Needle! In my Intro to Probability class, I did an equivalent experiment by dropping toothpicks onto a grid. I remember completing a big, complicated proof to show that it really was pi. Advanced note: If you are working your way towards an undergraduate math major (or maybe you already have one) can prove the relationship using a probability density function and double integrals.
If you want to know and/or try your hand at a simulated version of Buffon’s Needle: Science Friday did a write up on it:
This is a problem that you might consider at the beginning of a statistics course. Understanding how many combinations or permutations a certain set of objects can have is integral to doing probability and statistics. Unlike the odds in problem #1, the number of possibilities is really really big. So while you can’t truly “prove” that it has never existed before, the odds are ever in your favor.
So the next time you are stuck in the middle of a statistics or probability course and frustrated with the unintuitive nature of the problems. Don’t worry. The entire rest of the world is with you. And mathematicians have been thinking about this stuff for a long time. So while it may not feel intuitive, it’s definitely logical.