When Statistics Lie

All right statisticians, the gloves are off.  Get ready for a fight.

Home from college, my son presented me with the Monty Hall paradox (which I had encountered before with similar incredulity). With the self-assurance unique to denizens of the ivory tower, he argued passionately against my insistence that the universally accepted conclusion is a statistical fiction that has no basis in reality.

For the uninitiated, the famous problem goes like this:

You are a contestant on Let’s Make a Deal, and Monty Hall (the original show-host) offers you a choice of three doors. You choose Door Number 2. Obviously, your odds of winning the Ferrari are three-to-one against.

Monty then reveals that behind Door Number 3 is a goat. Not only are you still in the running, but your odds have just shortened to even-money.

So here’s the question: Given the option, should you stay with your original choice of Door Number 2 or switch your bet and take Door Number 1?

Most of us would say that it doesn’t matter. With two possibilities, your chances are 50-50, no matter which door you choose. So why switch?

Logical Nonsense

But that’s not what Statisticians say. Rather, since your original choice left you with a ⅔ chance of losing, one of the two ways you could have lost is now removed. Consequently, Door Number 1 now absorbs the ⅓ probability that previously resided with Door Number 3. In other words, the chance of the Ferrari appearing behind Door Number 2 remains at ⅓ while the chance of it appearing behind Door Number 1 doubles to ⅔.

Really?

Mathematically, this makes perfect sense. Practically speaking, it is utter nonsense. I’m still left with two unknowns, which are just as unknown as they were before the cranberry sauce appeared. Two chances: even-money; 50-50. That’s all there is to it.

No! Scream the statisticians. We’ve proven it mathematically. We’ve even tested it, and it works.

Well, maybe they have. I don’t know; I wasn’t there. But the popular illusionists Siegfried and Roy demonstrated a lot of interesting phenomena, too, so forgive me if a remain a skeptic.

You won’t forgive me, Mr. Statistician? Okay, I’ll prove I’m right.

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  1. #1 by dwachss on October 30, 2014 - 1:27 pm

    Rabbi Goldson: I’m sure you’ve heard from lots of statisticians, and I don’t dispute your final point about using statistics to lie, but you have misunderstood the Monty Hall paradox. The key to understanding why you should switch doors is the fact that Hall isn’t guessing where the goat is, the way you are; he’s looking behind the doors and knows. So when he tells you that one of the doors that you didn’t pick has a goat, then shows you a goat, he hasn’t told you anything you didn’t already know.

    Here’s the mental exercise to demonstrate that:

    I’m going to illustrate the situation where there’s a car (that you want) behind one door and nothing behind the other doors (to avoid the wise guy answer–http://xkcd.com/1282/)

    1. Hall offers you a choice of opening one door or opening two doors. You have a 2/3 chance of getting the car if you open two doors, so you pick that.

    2. Different scenario: You pick one door, then Hall offers you the choice of sticking to your door, or opening the other two doors. You have a 2/3 chance of winning if you switch.

    3. Same scenario as 2, but Hall adds: “I can look behind the doors, and I can see that only one of the two doors that you didn’t pick has a car”. You already knew that (there’s only one car), so that doesn’t change your decision. You should still switch.

    4. Same scenario as 3, but Hall then opens the door to the empty room (remember, he knows what door has what). Still doesn’t change your knowledge, or the probabilities. You still should switch.

    Scenario 4 is the Monty Hall paradox. The paradox arises from the apparent inconsistency between your choice initially and Hall’s perfect knowledge and reaction to your choice (if you initially picked the goat, then he wouldn’t show you that door).

    Any parallel to this paradox and Rabbi Akiva’s paradox of הכול צפוי, והרשות נתונה is left to you as the Rabbinic Authority.

    Danny Wachsstock

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