Say, when was the last time you deliberately sowed discord? Want to?
It’s easy, go find somebody close, like your spouse or sibling, and throw this riddle at them:
OK, so you’re on Let’s Make a Deal, and as always, you have to choose one door from a set of three, with a prize hidden behind one of the doors. Say you choose Door Number 1. Now, Monty Hall, as always, shows you that Door Number 2 has nothing behind it. That leaves Doors 1 and 3 still unopened, and the prize has to be behind one of them. Monty now gives you the chance to switch your choice, from 1 to 3. Should you do it?
The answer is unequivocally yes, every time.
This is where it usually gets ugly. Believe me, I know. I first heard this unaccountably incendiary little brain-teaser when my wife, Kathy, charged down the stairs brandishing a book titled The Curious Incident of the Dog in the Night-Time, in which the riddle and its answer are set out as part of the narrative. Why? I don’t know, it doesn’t matter, maybe because it illuminated some aspect of the narrator’s psyche, OK? Who cares? We’re getting off track. The point is, the riddle is in there, and so’s the answer, and as soon as Kathy read that answer she became almost violently agitated. She fairly bounded down to the living room, where I sat, little suspecting the poo storm that was about to hit me square in the face, and told me the riddle, told me the answer, and then demanded – demanded, mind you – that I refute it. “It’s wrong“, she insisted, “with only two doors the odds are 50-50 and there’s no point in switching!”
See, now, this is where your intelligent spouse, your spouse with a clue, would say “Hey! You’re right! Stupid book!”, but what do I do? Do I do the smart thing? Of course I do not. Nope, I just come out with “No, that’s right. You always switch”.
“No. It’s 50-50!”
“No, actually, it’s 66-33 that it’s behind the other door”.
“THERE ARE ONLY TWO POSSIBLE OUTCOMES. 50-50!!”
“Well, I could suggest that there’s a goat in the kitchen. There’s only two possibilities – there is a goat, or there isn’t a goat. Do you think the odds are 50-50 right now that there’s a goat in the kitchen?”
“YOU KNOW THERE ISN’T A GOAT”.
“Suppose I didn’t, and was just guessing?”
IT’S NOT THE SAME THING!”
“I’m just saying the possibility of only two outcomes doesn’t automatically mean the odds are 50-50…”
“WITH ONLY TWO DOORS LEFT AND ONE PRIZE IT’S 50-50!!!”
It went on for a bit like that, until I was a hair’s-breadth from getting a divorce. It’s probably as angry with me as Kathy’s ever been. If there’d been a stapler within reach, she would have rifled it at my head like she was Sandy Koufax and my face was the strike zone. As I kept trying to explain why the book was right, and she was wrong, knucklehead that I am, a change came over her. I could see it in her eyes. She wanted me to suffer. She wished anguish and spiritual torment upon me, just as I was visiting torment upon her.
There’s something about this deceptively upsetting little puzzler that drives people up the wall. You get the same reaction as if the riddle was “What’s bigger, an elephant or an ant?”, and you kept insisting that logically, it has to be the ant, it’s always the ant, ants are always bigger than even the biggest elephants. People want to take a swing at you.
Maybe not you, dear reader. Maybe you know the right answer is always to switch doors, and why it’s so, and don’t see why this riddle would vex anybody. On the other hand, maybe you agree with Kathy that it’s 50-50, so I’ll set out briefly why that’s wrong, but only because you’re not here to come at me with a a salad fork, determined to gouge out my eyes.
When you pick one door out of three, your odds of being right are, of course, 1/3, or roughly 33%. In effect you’ve created two sets of doors; a set of only one door, containing the one you picked, and a set containing the other two doors:
At this point, your door has a 33% chance of being the right one, and it’s 66% that it’s behind one of the other two doors.
So far so good, right? Now, and this is where it gets divisive, when Monty shows you that one of the two doors that you didn’t pick is empty, leaving only two possibilities for where the prize can be, it doesn’t change the odds that you were right in the first place. Those odds were frozen at the time you made your pick, and nothing now can change them – it’s still just 33% that you picked the right door. No, it is. It is. I’m telling you, it is. All you’ve done is change the composition of the two sets, there being only one door in each set. But the remaining door of the the unchosen two is now the only candidate in a set that was 66% likely to have the prize behind it. The whole 66% probability has now been collapsed down to a single door:
The prize never moved. You can’t improve the odds that your pick was right. It was one in three when you made it, and it still is, and knowing that there’s also an empty door among the unchosen pair doesn’t tell you anything new – of course there is. There’s only one prize. One of those other two doors has to be empty. No matter. The only way the odds could now shift to 50-50 would be to take the prize from where it already is and put it back in place by a random process. That hasn’t happened, so the fact remains, it was 66% that it was behind one of those two unchosen doors, and it still is, and now there’s only one door left to hold that probability. You should definitely switch then, because you’d be doubling your chances of getting the prize.
See? You see that, right? It certainly doesn’t make you angry, does it? If you were here, you certainly wouldn’t feel a diabolical urge to fastball a mug at my head, would you?
Hey, that’s another example of the 50-50 fallacy. There’s only two possibilities: either you do want to poke me in the eye with the business end of a pencil, or you don’t. Yet they’re not equally probable. Actually, I’d say the odds are better than 2-1 that you do.
After our heated debate, Kathy posed the riddle to all her friends, all of whom agreed there was no point in changing your choice because the odds were now 50-50, and all of whom also wished harm and misery upon me, when they were told I was insisting that the stupid book was right.
Kathy still thinks it’s 50-50, I think. There’s no way to be certain, really. I’m sure as shit not about to bring it up.