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?
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.
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.)
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.
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.
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.