h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

Calling all RMGs!

Please contribute to A2211832 in the Writing Workshop NOW!

I have a stupid question: how do I actually contribute? I've never written an article on H2G2 before.

Go to your space and find the bit that tells you you've written no entries (near the bottom). There should be a link there that says "Click here to add a new guide entry". Click it.

I have placed the glossary in the Writing Workshop. Add a contribution by posting it in the thread F57153?thread=405488&latest=1 and I will include suitable contents, and credit them.

Okay, but you should put it in the Collaborative Writing Workshop, which exists for precisely this sort of thing.

It has also been suggested (in the WW thread) that this should be in the Collaborative Workshop.

With apologies for my mistake, I shall remove this from there, and submit it to the Collaborative Workshop, as soon as I can find out how to do it. I hope all subscribers to this thread will follow!

New thread:

103?thread=10352/thread/406780" >F57152?thread=406780&latest=1

i like maths but my teacher is so boring
at the moment after school on a thursday i'm doing the criptic clues thing and i could end winning £10,000

how cool

Does being a statistician exclude me?

In a game, two (fair, six-sided) dice are rolled underneath a cover of some kind. The probability that their sum = 7, is 6/36, or 1/6. This corresponds to the six cells along the diagonal of the usual chart of outcomes for this situation.

Now suppose one person peeks beneath the cover and says, "there is a 2 showing". We want to know the probability that the dice add to 7, given that (at least) one of them is a 2. The formula for the conditional probability P(A|B) = P(AandB) / P(B) gives us that the probability that we now add up to 7 is (2/36) / (11/36) = 2/11. In the light of new information, the probabilty shifts.

Fine, except the same argument applies whichever number is reported to be showing, so suppose that someone coughs, and from where we're standing, we hear, "there is a -*cough*- showing". No matter what number, 2 or otherwise, was rendered inaudible by the cough, in any case the probabilty has climbed from 1/6 to 2/11, without any acual information being added.

What gives?

And, no, you're not excluded for being a statistician. I can just add you to the list, if you like.


GTB

It's very curious GTB, I've come across some similar ones but this is very strange. If you think about it as 'The probability that the two dice add to seven amongst all the possible dice rolls where a 2 is showing is 2/11' then this is perfectly believable because the 6 events 'a 1 is showing', 'a 2 is showing', etc. are not mutually exclusive. Not knowing how the person who peeks under the cover has decided what to say, this is the best information we have.

However, in the 'there is a *cough* showing' case we have no extra information. If A1 is 'there is a 1 showing', A2 is 'there is a 2 showing' etc. then AC='there is a *cough* showing' is the same as (A1 or A2 or A3 or A4 or A5 or A6). But P(AC)=1 and P('the dice add to 7'|AC)=P('the dice add to 7')=1/6.

Interestingly, if we knew something about how the person who peeks beneath the cover chose what to say, it would give us very different information. If the two dice were different colours, say red and blue, and he always said what the red dice was showing then the probability would still be 1/6. If he chose at random between the red and blue dice (50/50 which one he chose) and told us what that die was showing it would still be 1/6 (I think, haven't checked this). However, if he only told us that a 2 was showing if the sum didn't add up to 7, and otherwise said what the other die was showing then the probability would be 0. If he only says 'a 2 is showing' if either both the dice are showing 2 or a 2 is showing and the sum does add up to 7, then the probability is 10/11 I think.

I guess what makes us feel the 1/6 going to 2/11 is wrong is that we can easily imagine him choosing one or the other dice and reporting what is shows, or something similar, all of which give a probability of 1/6 rather than 2/11, but we find it quite difficult to imagine he has some perverse strategy like only saying a 2 is showing if they don't add up to 7.

Same paradox, simpler setup - you know that a certain woman has two children, with no information about their genders, so each of the four possiblities {(boy,boy),(boy,girl),(girl,boy),(girl,girl)} is equally likely.

Now you see the woman walking down the street with one of her children, a boy. What is the probability that the other child is a girl? From the above, we've eliminated the fourth option ((girl,girl)), so of the three remaining, two have the other child female, so the probability that the other child is a girl is 2/3.

Sneaky, eh?

You're on a game show where you're shown three doors. One has $1 billion behind it and the other have $1 each, and you win whatever's behind the door you pick. So you pick door #1. Then the announcer says, "It's a good thing you didn't pick door #2," and opens the door to reveal a one dollar bill. Then he offers you a chance to change your mind, i.e. pick door #3. Should you?

Well, the logic of the previous posts says yes. When you first picked door #1, there was a 1/3 chance of getting the billion dollars, which means that the previous two doors together have a probability of 2/3. Now that door #2 has been opened, that doesn't chance the chance of the prize being in 2 OR 3, but since 2 clearly doesn't have it, door #3 now has a probability of 2/3 of having the prize!

The flaw in the logic of all these, though (in my opinion), is the idea that probability is some fixed property of the game. It seems to me that probability is actually entirely based on our own knowledge of the object. In other words, if I flip a coin, catch it, and look at it (heads) while hiding it from you, then for me, the probability of it being heads is 100%, but as far as you're concerned, it's still 50/50. So in all of these puzzles, the assumption that the probability doesn't change, or that the same possibilities still exist with equal weights, just because we know something we didn't know before, is false.

In the case of my game show, your knowledge of what is behind door #2 simply changes the probability regarding the other two. There are now two doors, one with the billion dollar prize and one with a dollar bill, and the probability of the prize being in each is 50%.

In the case of the woman with the children, we have to break it down into four possibilities:
A) the boy you see is the elder, and the younger is also a boy,
B) the boy you see is the younger, and the elder is also a boy,
C) the boy you see is the elder, and the younger is a girl, or
D) the boy you see is the younger, and the elder is a girl.
Given this, there are two choices out of four that the other child is a girl, and thus the probability is 50%.

I'm too lazy to work out the numbers for the other one, but I would guess there's a similar argument for the dice.

Thanks for a fun problem!

Nonetheless, if you're on that gameshow again and again, and the same thing always happens to you then if you change your mind every time you'll end up twice as rich as if you stick with your initial choice every time. You can try it out with a friend, or do a computer simulation if you don't believe it.

And if you did a survey, walking up to random parents with one child with them, checking that they have precisely two children, and asking the gender of their other child, you'd find that two thirds of the time the gender of the other child will be different.

Okay, so what's the probability of this? Feeling bored, I clicked on "Infinite Improbability Drive" on the Front Page (it gives you a random edited Guide entry) and got this article: A543043
Then I started reading the conversations and came up with this thread: F66348?thread=118482

The arguments given in the thread seem pretty clear that I'm wrong, but I haven't been able to convince myself. Think I just need to sit down and crunch some numbers.

hey, 'tis EvilAl and i hope i'm interested in maths.. i've got a fair list of basic qualiications in it (showoff)

Hello. I don't know how active this place is right now. We'll have to wait, and see.

More active just now than it's been in months...

Welcome, EvilAl. I have added you to the dropdown on our homepage. That's about all you'll get by way of ceremony, but feel free to post mathy conversation, and someone'll probably talk back at ya.



GTB

Bobbyjoe656

Welcome; you're on the list!