Creating Passionate Users

Reese, Kevin, and The Monty Hall Problem

"There's a car behind one of those three doors... and goats behind the other two. Pick the door you think the car is behind. OK, now before I open the door you chose, I'll open one of the other doors... [door opens revealing a goat] and you can see there's no car behind that one. But I'll give you a chance to switch your door for the remaining closed door. What do you say? Do you want to switch or keep the one you originally chose?"

And that's one way of looking at the Monty Hall problem that's been inciting/confusing/intriguiging/pissing people off for years.
(You can try it yourself through this applet.)

In my recent blog on Seduction and Curiosity I made up a kind of variant on the Monty Hall problem, only this time the problem was for Kevin to pick the business card that had Reese's phone number on the back, from a pile of three business cards. After Kevin made his choice, Reese turned over a blank card, and asked if Kevin wanted to stick with the one he had, or switch for her remaining card. He declined, thinking, "Each card started with a 1-in-3 chance, and nothing could change that." But when he didn't switch, she criticized him for not realizing that switching would have improved his odds... something his friend Manny confirmed by saying, "Your odds would have gone from 1 in 3 to 2 in 3."

I wasn't really thinking when I posted the problem... it was just the first thing that came to me when I thought about the puzzles that have made me curious, so I used it as an example. (And you can definitely count me among the huge group of Those Who Did Not Get It when I first saw it.)

And I intentionally left a few things up for speculation that people have wondered and argued about in the Monty Hall problem including, "Does the motivation of the host matter?" "Do they ALWAYS offer the choice to switch, or does it depend on whether the host (Monty) knows that the contestant got lucky and picked the winning door..."

And then when I posted my Java code (in the comments to the original post), that repeated the Reese/Kevin thing 500 times, Daniel pointed out that there was the potential for another issue if my code was trying to model the real-life scenario of Reese and Kevin. The problem was that in my code Reese had a very specific algorithm--she always turned over the "A" card to show Kevin (revealing that it was blank) unless it happened to be the winning card, in which case she turned over the "C" card (Kevin always picked "B"). So that meant that if Kevin wasn't too drunk at this point, then he could eventually figure it out. Think about it... if Kevin is really paying attention, and he sees that most times Reese turns over the "A" card, then those times when she does turn over the "C" card, he thinks, "Oh... that must mean that the winning card is "A", because this time she didn't turn it over..." It would be just like if in the Monty Hall problem, Monty always looked at the doors from, say, left to right, and he always revealed the left-most of his two doors, unless that's where the car was. When the contestant saw him skip the left-most door and open the other one instead, the contestant would have new information he could use.

So, in my problem, the code was just meant to show you that if the EXACT same scenario were repeated 500 times, with Kevin always picking "B", and the winning card is randomly distributed among the three cards, and that if Reese ALWAYS reveals a blank card and offers Kevin the chance to switch, then indeed Kevin's chances go up to 2 in 3 instead of his original 1 in 3. In other words, in this example, Kevin should always switch. . (And this is exactly how Marilyn Vos Savant answered the question in the infamous article that started the controversy--she said the contestant should always switch.)

But as promised, here is my attempt at an explanation. But with

huge disclaimers:

1) I don't know or care where the math gene is... I was born without it. (Proving that good-at-programming doesn't always imply good-at-math). So, my explanation might be completely ridiculous.

2) This is not a subject I have ever--or will ever try to teach or write about, since it's WAY outside my knowledge comfort zone.

But if you're still reading and still wondering how switching could possibly improve the odds when all three business cards started with a 1 in 3 chance of being the right card... here's my biggest hint:

It's not about three individual cards. It's about two stacks.

Here's another way of thinking about it. Imagine I have, say, 50 cards divided into two stacks, and only one of the 50 cards has the winning phone number on the back. Stack X has just one card while stack Y has 49 cards. And now I pose the question, "Which stack do you want?" Hmmmm... which stack is more likely to hold the winning card? ; ) Of COURSE you choose stack Y, the big one. The odds of the winning card being in the bigger stack are 49 in 50. A hell of a lot better than the 1 in 50 odds you get if you pick the stack-of-one, stack X.

The point of the Monty Hall/Reese-and-Kevin thing is that it was always about two things, not three. It was always about the host's big stack (two doors/cards, etc.) vs. the contestant's small stack (one door/card).

The person running the show (Monty or Reese), starts out with a bigger stack!

Reese has the bigger stack... the one with a 2 in 3 chance instead of a 1 in 3 chance.
And by offering you the chance to switch, Reese is really giving you the chance to swap your SMALL stack for her BIG stack. It was always about Reese's bigger stack!

Her turning over a card made it look as though it came down to just the two cards--her remaining one and Kevin's original pick. But in reality, she was simply giving you the chance to switch for her bigger stack, and the fact that she revealed one of the cards in her stack doesn't change the fact that the winning card was always twice as likely to be in Reese's bigger stack.

But again, there's the important assumption in my explanation that says, "Reese always offers Kevin a chance to switch, no matter how many times they might repeat this." So Kevin doesn't have to wonder whether Reese is giving him a chance to switch only because she already knows he DID pick the winning card.

Finally, here's one more way to look at why switching ups the odds... the point is that if Kevin sticks to his original pick ("B") and doesn't switch, the only way he can win is if he got lucky the first time and "B" was the winning card--the 1 in 3 chance. But if he does switch, then he doubles his chances, because Reese always turns over a blank card. If the winning card is "A", Reese turns over "C" and gives Kevin a chance to pick "A". If the winning card is "C", Reese turns over "A" and gives Kevin a chance to pick "C". If Kevin always picks "B" and always switches, he has two chances to win instead of one. If Kevin always switches, then the only time he loses is when he happened to pick the winning card the first time (the 1 in 3 chance that "B" is the winning card). But if the winning card was either "A" or "C", then Kevin wins if he switches.

OK math heads and stats folks -- or anyone who has another way to think about this, please feel free to post your comments (and thanks so much to those who did in the original, with special mention to Brad Corbin, Matt Moran, and Woolstar) And if my explanation is just completely lame and makes no sense, feel free to say that too. I'm so out of my area of expertise here it's astonishing : )

And now, back to somewhat less geeky blogs...