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

http://www.bbc.co.uk/h2g2/guide/A525935

You know what the matrix is. It's all around us: it's everything we see, and everything we hear. You feel it when you go to work, and when you sleep. You can touch it when you pay your taxes, and...

*ahem* But this isn't about the matrix, it's about probability. Specifically, its about how, despite it being part of the fundamental basis of the universe via quantum mechanics, nobody really knows what it is.

Or something in that kinda "philosophy meets mathematics" vein. Comments, criticisms welcome. If you know what probability is, and can cause the entire entry to be made obsolete... stop off and give me advance warning on the way to collect your Nobel prize...

I think this is a another great one, Lucinda! My only suggestion would be to possibly change the title -- it led me to expect an entry that briefly explained what probability was, how to calculate probabilities, etc. Maybe "Philosophies of Probability" or something like that?


Mikey

I agree. The article should be entitled "Philosophy of Probability" or "Probability and Observation".

It is difficult to type words that are similar to word we type all the time without getting them wrong. In particular, it is obviously difficult for you to type "forehead" without it coming out as "format". You typed "the host has scribbled either a zero or a one on everyone's format".

The line "Most people, though, seem happy to not know what probability actually is" would sound better as "Most people, though, seem happy not knowing what probability actually is".

Good entry.

I'm guessing theres also a 'formalist' viewpoint on probability. That its just a mathematical formalism involving spaces of possible events measured in some way, and "hey, we just play about with it - if those crazy people we work for think it *means* something in the real world then thats their problem..."

Incidentally, best probability type puzzle I know of:

Puzzler: Suppose I've got 2 sealed envelopes, one of which contains twice as much money as the other. Here pick one and open it.
Victim: Ok...its got 100 pounds in it.
Puzzler: Right, suppose I give you the option to swap it for this other envelope I have, do you want to do it?
Victim: Well, its either twice or half, so either 200 or 50 pounds, so I either gain 100 or lose 50, and it must be 50/50 chance. Yeah lets go for it.
Puzzler: Ok, you want to swap. Now suppose there had been 1000 pounds in the envelope you first opened, would you still want to swap? (assuming you just want to maximise your winnings)
Victim: Well, either 2000 or 500 pounds, yeah I still want to swap.
Puzzler: So it doesn't matter how much money is in the first envelope, you still want to swap?
Victim: Yep.
Puzzler: Ok start again, 2 sealed envelopes, pick one *but don't open it*. Now it doesn't matter how much money is in the envelope you pick, so you don't need to open it, you still want to swap for this other envelope?
Victim: Er.. yeah...
Puzzler: Ok, have the other envelope, I'll take the first one back. Now, suppose I give you the option to swap it for this sealed envelope I have (that you've just given back to me), do you want to do it?
Victim: Er...
Puzzler: It contains either half or twice as much, just as before...

Where's the error in the logic?

HenryS, that is a wonderful probability paradox. I was going to mention the one with the three doors, but this one is better.

The 3 doors one (The Monty Hall Problem, to give it its proper name) is surprising (got me at first), but the correct answer is now (for me) intuitive. OTOH, the 2 envelopes problem still bothers me, I've yet to hear a really convincing explanation, though I know a few semi-convincing ones .

Nice entry, and it's on one of my fave topics !

I would include something about a formalist definition of probability, that a probability is a function p(x) with the properties (a) p(x) < 1 for all x and (b) (the integral from negative infinity to infinity of p(x)) = 1 - where 'x' is some random variable. With this definition, all of the math can be done without making any claims about how it applies to 'the real world,' whatever that is.

GTB

Okdoke, I'll do judicious cut and pasting for a formalist definition until I can find out more about it... How about "What do Probabilities Mean?" as a title? or "The Meaning of Probabilities"? format/forehead fix done: I've been writing my dissertation, so I've got computer stuff on the brain

The envelope swapping problem has a fairly simple solution, as follows: You say that the probability of the other envelope being double the size, as opposed to half the size, is equal, at 50/50.

Ok, so this means that the P($1) = P($4). Similarly, P($4) = P($16), etc. This chain (and similar chains) go on forever. If you then look at it, you find that the average amount of money in an envelope is infinite.

Every time you swap an envelope, your expected winnings is increased by 25%. But infinity times 1.25 is still infinity, so this isn't a problem.

Hmm - seems like a good topic for an entry - I wonder what its official name is? Monty Hall has entries in the guide around the place, but it looks like they're still under moderation: we had a big fight over it in misc chat if I recall. Nothing in edited though - so there's a collaborative effort that could happe there.

Yes, well you go as far as P($2^n) = P($2^(n+2)) for all n, then note that this means that your probability distribution has to be constant on an infinite range, which is impossible - there is no such probability distribution. Which really says that there isn't enough information given in the problem to determine the actual distribution, so you can't analyse it using expected winnings etc. But even without expectation in your toolkit, it still seems worthwhile swapping when you could either win 100 or lose 50. Thats the bit that bothers me.

The most authoritative resource I know for these problems is the rec.puzzles archive (should be easy enough to find on google given the url would be moderated), look under 'decision' section.

Another probability type problem, called the 'Sleeping Beauty Problem' might be worth including in a probability puzzle entry. The best resource for this can also be found by searching on google for 'sleeping beauty problems'.

It doesn't mean that there isn't enough info: it means that, as the problem is phrased, no such probability distribution is possible. {actually, it is possible: the limit as n tends to infinity of P(X) = 1/2n between -n and +n, and zero otherwise, is such a distribution. You can't express it as an analytical function, but it can still be a probability}

The double/half problem seems entirely reasonable - and swapping is certainly good, *provided* that the probabilities are exactly 50/50. In practice, that'll never be the case: you can look at the levels of prizes on previous editions of the show, or look at how much most game shows give away in prizes, and so forth. Knowing how much is in the first envelope gives you information: and if, using this information, you can deduce that the chances are 50/50 or better, you should swap.

Fuixes done. I've credited Henry and GTB for the formalism stuff, unless either wished to object...

Isn't that limit just zero everywhere? (and hence not a distribution since it doesn't integrate to 1) Hmm. Might depend on what your norm in the space is? If its not some sort of supremum norm but an L1 integration norm? Dunno.

Probabilities 50/50 - we don't know that. It just seems likely by symmetry and the fact that we don't know anything else. If we knew the distribution then knowing how much money is in the first envelope definitely makes a difference as to wether we want to swap, but the fact is that all we know is that one envelope contains twice as much as the other.

Ok, what am I missing? Envelope 1 (E1) contains x dollars. Envelope 2 (E2) contains .5x dollars with probability 1/2, and 2x dollars with probability 1/2. So the expected value of E1 is x, and the expected value of E2 is 1.25x. So you should switch, and then don't switch again, because E1 still contains only x dollars. The paradox depends on forgetting what the random variable x meant in the first place. You're saying that now, E1 contains either .5(1.25x) dollars or 2(1.25x) dollars, the expected value of which is 1.25(1.25x) dollars ( > x dollars!). But you know darn well that E1 contains x dollars, no more!

The correct way to calculate the probabilities of switching back is as follows: with probability of 1/2, E2 contains .5x dollars and I should switch back to gain .5x dollars back (value of switch-back: .5x). With probability of 1/2, E2 contains 2x dollars and I was right to switch in the first place; switching back will lose me the x dollars I just gained (value of switch-back: -x). So, the expected value of the switch-back is -.25x, so don't do it!

Does that make any sense to anyone else?

Excellent problem, btw!

GTB

Nice entry.

Small errors:
In the Subjectivism section (first paragraph), you missed an 'r': 'Unlike a fequentist...'
In the same section (last paragraph), there should probably be an apostrophe before the colon in "... people to 'put their money where their probabilities are: specifically ...".

Maybe call it "The schism of Probability"?

Oh dear... what a muddle I'm in now...

If the random variable x had been assigned to the value of E2 in the first place, then reverse everything in the above post...

Oh dear, I just don't know... Is this a mathematical proof of the statement "the grass is always greener on the other side"?

hmmmmmmmmmmmmmm....

Ok, how about this: one envelope contains x dollars, the other contains 2x dollars, right? When we pick E1, there is a 1/2 probabilty that we picked x, and a 1/2 probability that we picked 2x, giving E1 an expected value of .75x. The expected value of E2 is also .75x, making it neither advantageous nor disadvantageous to switch, which is an intuitively appealing answer, isn't it? I guess the paradox comes from some error in assigning the random variable in the first place, but I can't cite any particular principle that I saw violated...

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

BTW, I just read over the formalist bit in the entry (thx, Lucinda), and I think that condition 1 is wrong. p(x) > 1 is ok for individual x's (if x is a continuous variable), as long as condition 2 holds. A better first condition would be that p(x) >= 0 for all x. These two conditions would guarantee that 0 <= (the integral from a to b of p(x)dx) <= 1 (given a<=b, of course)

These conditions could be rewritten in English as 1: The probability of some event 'x' is always a non-negative number. 2: The probabilities of all possible (mutually exclusive) events total to one. For example, the probability of flipping 'heads' on a coin is 1/2, which is positive. The probability of flipping 'tails' is also 1/2, which is still positive. The total of these two probabilities is 1/2 + 1/2 = 1.

GTB

One possible solution to the envelope problem is that the value of money is proportional to the logarithm of the actual amount.

To a millionaire, a gain of £1000 is trivial. To a penniless tramp it is priceless. Similarly, if I find £1,000,000 in the first envelope, I would consider a drop to 500,000 to be at least as significant as a raise to 2,000,000.

If we define the value of an amount to be the logarithm to the base 10 of the actual money amount, then the value of E1 with £100 in it is 2. The value of E2 is either 2.3010 (in the case of £200) or 1.6990 (in the case of £50). The expected value of E2 is therefore the average of these which works out as 2 as well. There is thus no reason to change.

The same logarithmic definition of value is used instinctively by most contestants in Who Wants To Be A Millionaire, although they don't put it into Mathematical form.

And, of course, those who are already millionaires (such as on the celebrity shows) play "who wants to be a millionaire" a little differently -- while your average joe might decide to stick with 500,000 because that's too much money to risk, the celebrities inevitably keep going and going and going, even if they have no clue as to the answer. 2 reasons -- a) their level of wealth has desensitized them to the sensation of risking half a million dollars, and b) they don't keep the money anyway, it goes to charity.


Mikey

Editorial Note: This thread has been moved out of the Peer Review forum because this entry has now been recommended for the Edited Guide.

If they have not been along already, the Scout who recommended your entry will post here soon, to let you know what happens next. Meanwhile you can find out what will happen to your entry here: http://www.bbc.co.uk/h2g2/guide/SubEditors-Process

Congratulations!

Congrats! This one has been accepted! It now has the honor of being passed on to a sub-ed for editing, and then the further honor of passing through the hands of the Italics, and then the ultimate honor of appearing on the front page of the guide.


Mikey

There are some related details that the editors might like to consider adding. Probability was developed for the purpose of cheating at card games. It allows you to get useful information about the likely outcomes in specific situations. This only works for situations where the catagories are tightly defined (ie true or false, or 52 specific cards in a pack). If the catagories are fuzzy (ie how tall is tall), then you have to use fuzzy logic instead. the type of uncertainty that that probability is very good for is "uncertainty of outcome", where you know that it is definately one of the answers, but you don't know which. Also, lots of people get conditional probabilities wrong, as they just keep chaining new properties on the end of the chain. The answers only work properly if the properties only crop up in the chain once. When the properties crop up more than once, you rapidly start getting inacuracies due to having a P squared or P cubed term where you need the term to be P.
I don't know if this is all relavant, but I thought it better to add it than not.

I'm not sure use of probability in card games can really be called "cheating"...