Paradoxes

Well it's still only 9am, so I'm trying to think what else I can discuss. There was an interesting game that was brought up.  If you remember a while back when I introduced you to the prisoner's dilemma, and other game theory games, this one is similar, but not quite.  It's pretty simple, you start with a dollar in a basket, the host flips a coin, if it comes up heads the amount in the basket doubles, if it comes up tails you take whatever is in the basket and the game is over.  The question is how much would you be willing to pay to play this game?  50% of the time you'd win $1, 25% you'd win $2, and so on.  Each time the odds would half, but they'd never reach 0, thus there is no limit to what you could win.  Thus, unlike casino games where in the long run you'll always lose, for this game in the long run you'll always win.  No matter how much you pay to play each time eventually you'll win it all back.

I suppose a practice problem would arise similar to the old progressive betting system that you should be familiar with by now (double your bet each time you lose until you win), in that you couldn't afford to keep playing forever.  The question remains, how much would you pay to play?  Also looked at from the other side, how much would a casino have to charge so that it made a slight profit in the long run.  That's how you over come the problem of limited money supply.  If a casino offered this game, and it took $100 to play eventually someone would win back more money than anyone else had ever paid to play.  Still though according to the book most people would pay about $10-$20 to play it, and that seems about right.

So here is another more phlisopee question.  Now you are part of an experiment, it's pretty simple.  A doctor will give you a sleeping pill, and while you are asleep he will flip a coin, if the coin is heads he we wake you up and you leave.  But, if it's tails then he will wake you up and give you another sleeping pill.  The pill has a side effect though of making you forget what happened.  So every time you are woke up you can't remember any previous wake ups (if any).  Now, when the doctor wakes you he asks you what are the odds the coin landed tails up.  At first it seems obvious that it's still 50%, there no reason for the odds of coin toss to change.  But, think about it this way.  There are 3 possible times you would be woke up and asked that question, out of those 3 times 1 would happen when the coin was heads, and 2 would happen when it was tails.  Thus 66% the coin would have be tails up.

Here's a more extreme example.  Instead of a coin toss it's a wheel, with a 1 in 100 chance of coming up a particular way (A note, I'll be using the term 'the wheel comes up' to mean it comes up with a particular number, and it had a 1 out of a 100 chance of doing that).  If it does come up that way then you will be put back to sleep, over and over, 10,000 times (the wheel is only spun once), if it dosen't you will be woken up and walk away.  So there is a 99% chance you will go to sleep once, and then wake up and leave, but 1% of the time you will go to sleep and wake up 10,000 times a row and not know it.  Now think of it from the doctors point of view.  If he were to ask 'do you think the wheel came up or not?', it would seem that by answering no 99% of his patients would be right.  However, most of the times he asks that question the correct answer is yes (since he does it 10,000 times each time it happens).  Going farther still, say he only wakes you up at all when it comes up, if it dosen't he kills you, or puts you in a coma (or wakes you with out asking you the question).  Now it should be obvious that you must answer yes it did come up, since that is the only way you could have arrived in that situation.

I just thought of another way to look at it, with out all the crazy sleeping pill nonsense.  Say you take a poll, you flip a coin and then ask people if they thought it was heads or tails.  The odds would be 50% they'd be right. However, if you told them that whenever the coin came up tails you only asked one person, but when it came up heads you asked 100 people (for that single toss, and none of them know about the other people getting asked). Then that means for most of the people who you are asking, the answer is heads.  Going back to our wheel, you tell them that if it comes up (1 in 100 chance) then you ask 10,000 people, but when it doesn't (99%) you only ask 1.  After spinning the wheel 100 times, it should have come up once, and didn't 99 times.  Thus, you should have asked 99 people the question when the right answer was no, but 10,000 people when the right answer was yes. Again most people you ask would be right to guess that the wheel had come up.  Now put your self in the position of a person being asked, and also put some money on the line, say $10,000 if you correctly guess if the wheel had come up or not (if you guess right you win the money, even if you guess the wheel didn't come up [99 out of 100] and it didn't).  Would you guess it had come up or not?

I would guess it had, every time.  At first this is pretty crazy, but it seems fairly obvious with that last example with different people, I just hope I didn't miss some point of logic that fundamentally changes the game.

http://en.wikipedia.org/wiki/Sleeping_Beauty_problem

Next up on my tour of Probability theory paradoxes we will visit the common Monty Hall Problem.  This is pretty often told, so there is a high probability you've heard it, but either way here it is again (briefly).  You are on a game show, there are 3 doors, one of which has a prize behind it. You pick a door and if you pick the door with the prize you win.  Here's the twist, after you pick, the host will open one of the other prizeless doors, and then give you the option to change doors.  The question is is it better for you to switch, or does it make no difference?  At first it would seem to no matter, they are all 33%, and even after one is opened at most it's now 50/50, but both doors still have the same chances.  However, here is the way that I understood it.  When the host opens a door he must open one without a prize, and one that you didn't pick.  There are two scenarios, either you picked the right door at first (33%), and thus he has two doors without prizes that you didn't pick to open one of.  Thus, if you change your door you will change to one with out a prize, but if you keep with your orginal you will win.  Thus for the 33% of the time you picked the correct door at the start you will always win to stick with your pick, and always lose to switch.

Now what is the other possibility?  That you picked the wrong door at the start (66%), now the host dosen't have a choice, he can only open the other prizeless door.  He can't open the one you picked, and he can't open the one with the prize.  Now the only doors left are yours (prizeless) and the other (with prize).  Thus, 66% when you start with the wrong door you will always win by switching, and always lose by staying.  So it should be clear that 66% you will win if you switch, but you will only win 33% by always sticking.

http://en.wikipedia.org/wiki/Monty_Hall_problem

http://en.wikipedia.org/wiki/Category:Probability_theory_paradoxes

Now for my next act.  This one is actually pretty interesting.  It's less visual, but whatever.  When you have a body of statistics broken up in some way, by combining them you can change leading statistic.  It's much easier to give an example:
                    1995                    1996            Combined
Derek Jeter            12/48 .250                183/582 .314    195/630 .310
David Justice        104/411 .253            45/140 .321        149/551 .270
Now, in case the spacing on that terrible tabbed chart dosen't work out, the bottom guy has the better batting average in 95 and 96, but the combined total the top guy is better.  This seems like a paradox, how could he be better each year, but the average still be worst?  And in that case who is better?  If you look at the numbers you can figure it out.  In 95 they both didn't do well, but davey did slightly better, however derek barely played in that year.  I don't know what the second number is, but I assume it's attempts to hit the ball, and the first is sucesses.  So in 95 derek only tried 48 times, where davey tried 411 times.  As you should know the larger a sample you have the more accurate your statitics are, so the 95 statistic is much more accurate for davey than for derek.  Now we move to 96.  In this year the opposite is true, they both did good, but derek has a lot more samples, and thus his statistic of doing good is more meaningful than davey's.

To take this to the extream, say that in 95 derek only tried once, and failed, but davey tried 100 times, and only hit it 10 times.  For 95 derek's average would be 0.000, but davey's would be .100, at the same time davey's would be much more accurate picture of his actual skill, where as derek's would be a fluke.  Now in 96 derek tried 100 times and hit the ball 90 times, where as davey only tired once, and hit it.  Now for 96 derek's average would be .900, and davey's would be 1.000.  So again in both years davey would out perform derek, but taken as a whole derek would have 90/101 = .891, and davey would only have 11/101 = .109.  Thus with more data we see a much more accurate picture.

You can see how if I were trying to convice some one of davey's baseball skill I could use this to my advantage.  You can almost always make a statistic to say just about anything you want to convice people of.  This is where that famous Mark Twain quote comes from (paraphrased), there are 3 types of lies: lies, damned lies, and statistics.

http://en.wikipedia.org/wiki/Simpson%27s_paradox