I've been meaning to write about Monty Hall for ages. This is the well-worn probability problem based on a game show. There are three closed doors, with a car hidden behind one of them, and goats behind the other two. The contestant chooses a door, and the game show host then opens a different door, which has a goat behind it. The contestant is then offered the choice to swap to the other door, or stick with the original one. If the contestant settles on the door with the car behind it, they win it.
The obvious question is, should they swap, and what is the probability of winning if they do/do not swap? This problem has been published in magazines, resulting in decades of correspondence and controversy (ok, not decades, probably).
The "wrong" answer (and the scare quotes are appropriate) is that once one door has been opened, the prize is equiprobably behind the other two and it doesn't make any difference if the contestant swaps or not. The "correct" answer is that they should swap, as the probabilities are easily calculated to be 2/3 and 1/3 respectively.
The real answer is that it depends. In particular, it depends on the problem being very precisely stated, or alternatively, it depends on the assumptions that people make in the absence of a clear statement. As I wrote it above, it's actually a little unclear, though I wouldn't criticise anyone who made the obvious assumptions and who came up with the "right" answer. Wikipedia has a decent discussion of the ambiguities and history of the problem.
Notably, even when the problem is precisely specified, a large majority of people still get it wrong. Which only goes to show that people, even supposedly intelligent and educated ones, can still have a complete mental block where probability is concerned.
Another popular problem is the following: a person has two children, at least one of which is a boy. What is the probability that the other child is also a boy? Again, there is a wrong (but common) answer of 1/2, and a "right" answer of 1/3. Again, the problem is typically ambiguous in its statement. A more interesting version of it appeared on blogs and the "More or less" BBC Radio4 program recently: a person has two children, at least one of which is a boy born on a Tuesday. What is the probability that the other child is also a boy?
To give you the fun of thinking about it yourselves, I'll not not include any links and continue this in a later post...
And the title? I couldn't resist the juxtaposition of the fact that someone thought it was worth writing a whole book about this simple maths problem that pigeons can solve!
The obvious question is, should they swap, and what is the probability of winning if they do/do not swap? This problem has been published in magazines, resulting in decades of correspondence and controversy (ok, not decades, probably).
The "wrong" answer (and the scare quotes are appropriate) is that once one door has been opened, the prize is equiprobably behind the other two and it doesn't make any difference if the contestant swaps or not. The "correct" answer is that they should swap, as the probabilities are easily calculated to be 2/3 and 1/3 respectively.
The real answer is that it depends. In particular, it depends on the problem being very precisely stated, or alternatively, it depends on the assumptions that people make in the absence of a clear statement. As I wrote it above, it's actually a little unclear, though I wouldn't criticise anyone who made the obvious assumptions and who came up with the "right" answer. Wikipedia has a decent discussion of the ambiguities and history of the problem.
Notably, even when the problem is precisely specified, a large majority of people still get it wrong. Which only goes to show that people, even supposedly intelligent and educated ones, can still have a complete mental block where probability is concerned.
Another popular problem is the following: a person has two children, at least one of which is a boy. What is the probability that the other child is also a boy? Again, there is a wrong (but common) answer of 1/2, and a "right" answer of 1/3. Again, the problem is typically ambiguous in its statement. A more interesting version of it appeared on blogs and the "More or less" BBC Radio4 program recently: a person has two children, at least one of which is a boy born on a Tuesday. What is the probability that the other child is also a boy?
To give you the fun of thinking about it yourselves, I'll not not include any links and continue this in a later post...
And the title? I couldn't resist the juxtaposition of the fact that someone thought it was worth writing a whole book about this simple maths problem that pigeons can solve!
11 comments:
The pigeon part reminds me of a discussion on (I think) Material World on R4 where they compared the left & right side of human brains and rats. The idea was a random "pattern" - the left side tried to find a pattern and the right side & rats saw no pattern )or didn't try).
The rat example was some doors, where food would appear randomly. The rats would just sit outside one door waiting for the food and ignoring the other doors. They did better than people who tried to spot a pattern and go to the correct "door". I can't remember how they separated left & right side of brain, but the memorable quote was "The right side of our brain is just like a big rat..."
Forgot to say that the dilemma's easy for me, as I'd select the door the host opened, as I'd prefer the goat.
But IIRC some pigeons are Russian astronomers with a tendency to get the wrong answer. That has to be good for the dullest of sciencebloggers.
Skanky, that's memorable enough to be googlable, as well....seems to be this.
That's the one, thanks. I hadn't tried, because I'd assumed I'd listened to it. But as soon as I saw that, I realised it had been linked to from the MindHacks site:
http://mindhacks.com/
Which I have lost the habit of following - thank Guha for RSS. :)
Anyway, seems I'd misremembered it - which is par for the course.
An interesting thought is whether Monty had worked this out before they came up with the game
Birds are quite intellegent.
Even pigeons, I suppose.
I don't really get why the monty hall problem is so hard for people.
The host gives information when he opens one of the doors.
One could even think about oneself actually planning to switch the choice as soon as the door is opened.
But the two children issue, well, that's always been an unresolved paradox for me (it was introduced as two parrots in a cage, and the owner tells that one of them is male... what if he told that the black one is male instead?), how come the odds change if the information given is entirely useless? It's probably a scope issue.
The weekday twist highlights that.
I'll demonstrate my sub-pigeonness now, hope this motivates you to write a post explaining what's the issue:
Let's start narrowing it down.
(7 weekdays * 2 genders)^2 = 196 combinations. 49 boy-boys, 49-girl-girls and 98 girl-boys (order insignificant). Only boy-boys and girl-boys here so 147. 98/147 odds the other is a girl at this point. or 2/3.
And then, the boy is born on tuesday so divide by seven the number of boy-boy combinations, so we have 7 and 98. So 7/105 that the other child is a boy and 98/105 that the other child is a girl.
Ouch, that must be wrong. The assumptions are not well justified here. I'll leave it to the expert to illuminate... :)
Eräs,
I think your error is just a little carelessness, not a big conceptual problem. Of the 49 BB pairs, actually 13 have a Tuesday birth (7 first-born, 7 second-born minus the case where both are Tuesday which has been double-counted). For the 98 BG pairs (order unimportant) you still have to divide by 7 there too!
Followup post is here.
Ouch, that was a strange asymmetry error to make... and the other wasn't nice either. Shows what you get for posting in the middle of the night...
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