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# When Statistics Lie

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.

## What comes next?

### A carni at the state fair flips a silver dollar nine times in a row, and it comes up heads every time. What are the odds on the next toss? (See answer below.)

## Win-win?

Let’s tweak the scenario and say there are three contestants: Larry, Moe, and Curly. Larry picks Door Number 1; Moe picks Door Number 2; Curly picks Door Number 3 and gets the goat, so now it’s just Larry and Moe in contention for the Ferrari.

A thought pops into Larry’s head at the same moment it pops into Moe’s. Each says to himself: Wait! I’ve heard of this. It’s the *Monty Hall paradox*. Eagerly, each agrees to trade his Door for the other’s.

So now, according to statistics, Larry has a 67% chance to win and Moe has a 67% chance to win, giving them a combined probability of 134% to win. Needless to say, the probability of winning cannot exceed 100%. Or maybe the Ferrari will grow a back seat.

No doubt statisticians have an answer. I’m eager to hear it.

## Excerpts from the Famous Fallacies of Charles Lamb

**Crime never pays:** But the rogues of this world -- the prudenter part of them, at least -- know better; and, if the observation had been as true as it is old, would not have failed by this time to have discovered it.

**Don’t laugh at your own joke:** The severest exaction surely ever invented upon the self-denial of poor human nature! This is to expect a gentleman to give a treat without partaking of it; to sit esurient at his own table, and commend the flavour of his venison upon the absurd strength of his never touching it himself.

**A bully is always a coward:** Pretensions do not uniformly bespeak non-performance. A modest inoffensive deportment does not necessarily imply valour; neither does the absence of it justify us in denying that quality.

**When two argue, the warmer is generally in the wrong:** Our experience would lead us to quite an opposite conclusion. Temper, indeed, is no test of truth; but warmth and earnestness are a proof at least of a man's own conviction of the rectitude of that which he maintains. Coolness is as often the result of an unprincipled indifference to truth or falsehood, as of a sober confidence in a man's own side in a dispute. Nothing is more insulting sometimes than the appearance of this philosophic temper.

**If you love me, you must love my dog:** The good things of life are not to be had singly, but come to us with a mixture; like a schoolboy's holiday, with a task affixed to the tail of it.

## Paradoxical Proof

The problem with abstract thinking is that it doesn't always carry over into the real world. Hypotheses and theoretical constructs produce many wonderful innovations, and thinking outside the box is what moves mankind forward in countless different ways.

But simple statistics can produce misleading results by ignoring causation or various other factors. Abelson’s paradox demonstrates that batting averages are virtually meaningless when predicting success for any given at-bat. Simpson’s paradox shows how trends in different groups of data can reverse themselves when the same data groups are combined. And Abraham Wald determined, during World War II, that navy bombers could be made safer by observing the pattern of bullet holes on returning planes and reinforcing the areas where the planes were *undamaged*.

Nowhere are statistics less reliable than in human psychology and public opinion. Take voter polling. A week before the 2014 midterm elections, 68% of voters assert that the country is on the wrong track, 72% describe the economy as weak, and 62% lack confidence in President Obama’s leadership. Even so, paradoxically, a majority still favor the Democratic Party.

This might be a good place to mention that, according to one poll, 11% of self-described atheists claim to believe in God.

## Answer to question above.

100% heads. In statistics, this question should always begin with the phrase, "assume a fair coin." If so, the odds remain at 50-50. If not, it's a near-certain bet that the coin has heads on both sides.

## Delicate Balance

After creativity soars, it always has to come back to earth. Only by striking the delicate balance between imagination and reality can we realize the benefits of both. As the late Senator Daniel Patrick Moynihan famously said, “You’re entitled to your own opinion, but you’re not entitled to your own facts.”

Or, in other words, perpetual motion is a wonderful idea, but it won’t run your Ferrari.

## Comments

As you described it, "Monty then reveals that behind Door Number 3 is a goat." That's ambiguous. Why did Monty open door number 3? If he picked it at random, then showing a goat gives you new information and the change of winning is now 1/2. If he did it the way I described the "paradox" then the chances have not changed. You have to know the rules in order to calculate the probabilities. That's the problem with your "Larry, Curly and Moe" counter example. How did Monty select Moe's door to open? (I can go into this more if you want).

That's exactly your point with the heads-nine-times-in-a-row example.

So I agree with your overall argument that pushing math into an argument can be misleading, since real life never is as cleanly defined as a game is. However, if you're playing Let's Make a Deal and can assume they're being fair with you, go ahead and trust the mathematicians.

Danny

The point is that Monty Hall's knowledge doesn't change the odds. He's giving you a choice of opening one door or opening two doors, then telling you "one of two doors has a goat".

He's not picking a door at random and lo and behold--it's a goat! Now you know there's a 50% chance of the car behind the other two doors. He's deliberately choosing to open a door that gives you no new information.

Moshe Koppel (professor at Bar Ilan) had a nice article in Tradition 37:1 on the application (or lack thereof) of probability theory to halacha (rov and safek).

Danny

"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."

No you should not. Doesn't matter either way.

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

"So now, according to statistics, Larry has a 67% chance to win and Moe has a 67% chance to win, giving them a combined probability of 134% to win."

No, they both have a 50% chance. It still adds up to 100%.

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