Random Rambling: Math

We have... let's say an eclectic mix of interests, and we feel our patrons do too.  With that thought in mind, we present a feature we call "Random Rambling."  It isn't quite random, but it's close:  we made a little spinner with some of our favorite subjects.  Every Friday, we give the thing a spin, and then you, lucky people that you are, get to learn a fact related to the subject the spinner landed on.

This week's topic is:

Math!

In that case, why don't we talk about goats?

The world of math, just like any world populated by people, is rife with legends, both normal and "urban."  Case in point:  The Monty Hall Problem.

The Monty Hall Problem

Back in the 1960s (more than 60 years ago now), a game show began production called "Let's Make a Deal" hosted by the affable Monty Hall.  It was a nominally popular variety game show, with a bunch of different "trades" offered to contestants.  One of the trades in particular caught the interest of mathematicians due to a peculiar quirk in probabilities.  The trade would work as follows:

1. A contestant, often dressed absurdly, is picked out of the audience.  They are then presented with three doors.  Behind two doors, they are told, is a goat.  Behind the third door is a car!
2. The contestant gets to choose one door.  For the math we're about to do, the door is assumed to be chosen at random; they don't get to listen at each door to see if there are any bleats on the other side.
3. No matter which door they choose, Monty Hall then revealed what was behind one of the other two doors.  This was, invariably, a goat.
4. Monty would then ask the contestant whether they wanted to stay with the door they had or switch to the remaining door.

Mathematicians asked themselves asked whether it was better to stay with the original door or switch.  They did the math, and promptly tore out their hair.  Why?

Because, despite how it might look, you were twice as likely to win the car if you switched.

The problem with the math is that the probability you are experiencing is not the same as the probability you are perceiving.  Let's talk this out.  If the door you choose is random, then when you first choose your door, you have a 1 in 3 chance of selecting the door with the car.  This makes sense, right?

Then Monty removes a door.  You are now left with two doors:  the one you originally chose and the one you can switch to.  That's a 1 in 2 chance to pick the correct door, no matter which of the two you pick!  So how does it make sense that the other door is twice as likely to have the car?

Well, because you've made the mistake of thinking that there are only two possible solutions.  There are actually six.

You stay at Door X	You switch to New Door
Car was at Door X:  You win a car	Car was at Door X:  You win a goat
Car was at Door Y:  You win a goat	Car wasn't at Door X:  You win a car
Car was at Door Z:  You win a goat	Car wasn't at Door X:  You win a car

What trips people up is that pair of solutions in the lower right.  You see, you have to account for two different possibilities, but they both involve the exact same thing happening:  you picked a door that didn't have the car, Monty Hall removed a door, and you picked the last door.  Whether that door was Y or Z didn't matter, because every single time, Monty Hall removed a door with a goat.  Because of that, the only options you have to worry about are whether you originally chose the right door (a 1 in 3 chance) or the wrong door (a 2 in 3 chance).  If you chose the wrong door and then switch, you're guaranteed a win, since the only door left is the one with the car.

That's the non-intuitive part.  Removing the door and offering you a deal doesn't change the odds that you picked the right door initially.  When Monty Hall offers you the deal to switch doors, he's effectively asking you, "Did you pick the door with the car?  Guess correctly and I'll give you a car."

The Monty Hall Problem Problem

So, funny story.  The deal we just talked about?  The one presented by Monty Hall on "Let's Make a Deal"?

It never happened.

What actually happened was that a reader sent a question to Marilyn vos Savant, a lovely lady with the Guinness World Record for the highest IQ score.  This question, in turn, was derived from a question (and answer) posed by Steve Selvin way back in 1975.  Selvin, when other mathematicians inevitably decided he couldn't possibly be right, wrote a short paper explaining how the math worked.  The paper was called On the Monty Hall Problem, so called because the scenario in question reminded him of the game show.

The Monty Hall Problem is the victim of a game of "telephone," an urban legend just as much as those tales repeated as truth, with a source of "my cousin's brother's roommate knew the guy."  Originally a weird bit of math that reminded someone of a game show, it became conflated over time until, if someone tries to explain it to you, it is invariably presented as an actual deal on the original show.

Conclusion

Math is sometimes presented as a universal truth.  It isn't.  Math, just like real life, is prone to lies, and strangeness, and even legends and fables.  That part about it being universal and true is more of a reputation thing than anything else.  If you want actual, universal truth, you should check back here every Friday.

After all, would we lie to you?