Yep, I’ve run simulations on it to show that switching works. But why it works is harder to articulate. jhbdager has said the same thing, I see.
The hardest thing to learn about statistics and probability: never trust your gut.
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Even Marylin Vos Savant had trouble convincing phds of the truth.
EDIT: Before I said mathematicians, but a closer reading of Ms. Vos Savant’s column shows that she did not out any of the learned people berating her as being from a particular academic discipline.
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I hate feeling stupid. I want to understand this. But the jelly bean problem doesn’t seem to have anything to it, after I choose the green one. What happens next?
Then you follow the link to the original article and get pop-ups thrown at you to hear the rest of it.
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Argh! Usually they put the link to the full article at the bottom of the page, where I’ve come to expect it. the “wait but why” link doesn’t look at all like what you’d click to get the rest of the article, it looks like what you’d click if you wanted deeper background on the author. Puzzles within puzzles, I guess.
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The puzzle I find myself trying to solve, isn’t why switching improves my odds, but rather, what am I missing that would make this intuitively obvious?
The gimmick that I can intuitively grasp, is that the jelly bean I’m holding is not available for him to snatch up and call poison. If it were one of the options, then the odds would be even (so to speak) and he wouldn’t be giving me any new information. If I had taken two jelly beans, I’d have a 100% chance of dying, but he’d have a 1 in 3 chance of being unable to pull away a poisoned candy.
So now I’m thinking that one through. I grab two, and 2/6ths of the time, he says, “oops, I can’t remove a poisoned bean”, and I know to swap both of them for the one on the stump. The other 4/6ths of the time, he pulls the one from the stump, and I pull one out of my hand, and I’ve got a 2/6ths chance of getting that guess right, and somehow all those 6ths add up inutively to 6 out of 6.
The other mental glitch in understanding this puzzle, is the idea that this is a repeatable statistics exercise, but I personally only get to lose it once. To properly appreciate it, I have to imagine I’m a character from To your scattered bodies go, or The Prestige.
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I find it incredibly intuitive when you diagram it out:
- move strategy
- Pick loser A, (reveals B) move == win
- Pick loser B, (reveals A) move == win
- Pick winner (reveals A or B), move == loss
- stay strategy
- Pick loser A (reveals B), stay == loss
- Pick loser B (reveals A), stay == loss
- Pick winner (reveals A or B), stay == win
So, if you move… then you have a 2/3 chance of winning. Staying has a 1/3 chance.
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It’s easier with a random number generator handy.
“Say I pick door 1. If it is door 1, I win by not switching, otherwise I win by switching. Now let’s generate 10,000 random numbers between 1 and 3.”
I’m also a fan of the thousand door version.
Over the run of Money Hall’s show, I wonder how many times the contestants took that leap and got the car? How many times did they stay put and get the goat? That’s no longer a simulation, but real empirical data.
I pick a jelly bean. I have a 1/3 chance of having picked the right one.
Nothing my executioner does afterwards can possibly change this fact, as it would require time travel.
I find that part alone pretty convincing.
Once you realize that there is no time machine involved in this problem, you should go from “I still need to be convinced why I should switch” to “I still need to understand where my intuition went wrong”.
Now, after I’ve picked one jelly bean, I know that I probably picked the wrong one. But switching to a different one wouldn’t help, because I don’t know what bean to switch to.
Next, the executioner tells me which of the other two is not poisonous. That is, in the third of cases where I did pick correctly in the first place, he just chooses a random bean and tells me nothing. But if I picked wrongly in the first place, he tells me which of the other beans I should switch to if I picked wrongly in the first place. Exactly the information I was looking for.
The 100-bean version (I find 1000 excessive) makes it clearer as apparently, the numbers 1/3 and 1/2 are so close together that they are too easy to confuse.
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Yeah, somebody’s had to collect that data by now (assuming films/tapes of the show survive, which isn’t always the case for pre 1980s TV). Google doesn’t seem to find the data, though.
My brother was trying to explain this problem to my father who just wouldn’t accept the answer, so they got out a deck of cards and actually just played the game. My father shuffled three cards and my brother picked one, looking for the queen, he always switched.
My brother got the queen 26 times in a row.
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The probability of correctly winning a bet with 2/3 chance of success, 26 times in a row, is 1 in 37877. Seemingly a quite remarkable outcome. But maybe hundreds of thousands of people reading this blog did something similar and only the ones with an unlikely outcome (like yourself) posted about it.
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A friend of mine in university used to say, “If you want a one in a million chance to happen, just look around, they happen everywhere. If you want a particular one in a million chance to happen, you’ll be waiting a long time.”
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Exactly. Amazing how many people don’t get that, though.
Clearly the real problem is this vigilante plum farmer, meting out punishment disproportionate to the crime. The only solutions are escape and/or self-defense.
I love this effect—I’ve heard it called the “blade of grass” paradox.
I can’t wait to finish my stats class, because then maybe I’ll understand how to do this stuff.
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For a really tough audience, 1000 doors can be insufficient. Often you have to up it to 105 or 106 doors. In an extreme case, I had to up it to 10n, for arbitrarily large values of n.
Is that you, Vizzini?
Edit: I see that Malarkey beat me to the reference.
