## Tuesday, December 09, 2003

love that one...
mathematically it will always be best for the player to redo (undo) his selection when Monty has opened a door (it gives you 66.6666....7% chance to win)...

In the beginning the player has a 33.33% chance of choosing the right door and there is 66.67% chance of the big prize being in another door.
The player knows that Monty is going to open a door that does NOT contain the big prize. So when he does, this should not affect his belief of his door being the right one. He still has a 33.33% chance of having selected the right door in the FIRST place. Thus there must be 66.67% chance of the big prize being somewhere else (behind the last remaining door)...

I'd like to see something a little more human pop up in a later reading list because one aspect of Max Payne that significantly added to the game was added as an afterthought rather than something that any amount of business management courses, books, or money could predict.

And that's where I think we part company. The world is full of perfectly marketed and managed products to the point of overkill. Your book recommendations, while perfectly good, are all concerned with the what of success. You never seem to touch on the why.

"So, if you're that contestant, do you stick with the door you originally picked, or do you switch?"

Stick. It's a probability problem. The first time around you have a 50 percent chance of being right, and the second time around you have a much lower percent chance of being right. I'll leave the exact figure for someone else to figure out.

"So, if you're that contestant, do you stick with the door you originally picked, or do you switch? "

Definitely change. The first door you choose has a 33% chance of being right. Monty opens the TV-door, and you're still stuck with a 33% chance of being right. I forget if this is filed under 'odds' or 'probability', (probability I believe?) but if you switch you've got a 67% chance at being right.

Imagine if he opened the car door first. Just sit down and cry. The guy's just screwing with your mind. ;)

Hey, Charles. I see three explanations here. Two of them are right. The other one is, well, yours. Care to stick or switch? ;)

Yeah. Always switch. The two explanations given are entirely correct, but of course even university math professors have written in to vos Savant, adamantly scolding her for being so foolish. :)

Here's the classical thought-experiment that usually helps clear it all for non nonbelievers:

This time there are a MILLION doors, and just one prize. You pick a door, and then KAPOW! Monty hurls open 999,998 of the other doors, leaving door #36,971 suspiciously closed. NOW do you stick with your original choice, or switch?

I don't care if I'm wrong. It's MY door. :P

Winning and losing are both opportunties even if the norm of typical thinking likes to condition us differently. That's what bothers me about Scotts reading list. It knows the cost of everything and the value of nothing. I'd rather see an exhibition of Edo glass than read any one of those books, excellent suggestions though they are.

Now, which door are you going to pick? :)

Okay, we got some smart guys here who've already spoiled the fun. Switching is always recommended. Jay, funny you use that million door example, as that's the same example I use when trying to convince people who keep insisting that it's a 50/50 pick either way, so why bother switching.

In 1995, back when Rise of the Tried was being made here (one of "Apogee's" last games), I presented this puzzle to the three coders, all highly skilled in math, and none could be convinced that you should switch. Yet, among themselves they disagreed, with one thinking it was 50/50, one thinking it remained a one-third chance, even if you switched, and the other thinking that by staying your odds jumped to two-thirds!

Another good way to show people is to use a quarter (the car), a dime (the TV) and a penny (the donkey). The first time you pick the quarter, remove either the dime or the penny, and switching proves to be a mistake. But, when you pick the dime or the penny first, switching earns you the car (because the quarter is never removed). So, this shows that switching wins two thirds of the time.

Here's another counter-intuitive puzzle:

A couple has two kids, and they happen to mention that one of them is a girl and plays soccer. What are the odds that their other kid is also a girl?

-- "I'd like to see something a little more human pop up in a later reading list because one aspect of Max Payne that significantly added to the game was added as an afterthought rather than something that any amount of business management courses, books, or money could predict...Your book recommendations, while perfectly good, are all concerned with the what of success. You never seem to touch on the why."

Charles, not really sure what you're asking here?

Aw, heck. I heard different. Never mind.

"Charles, not really sure what you're asking here?"

You've made a lot of excellent comments and suggestions that many people will find useful, though I find there's areas you don't readily touch on such as people, culture, and history. The point being that to create something, as opposed to grinding though a mechanical process, requires a little bottom, as Winston Churchill might say.

Perhaps I'm being a little difficult here or the tide of history is rolling over me. Either way I've seen a lot of people, goods, and services delivered badly because the people behind them are box tickers. They know everything and understand nothing. Books on value free (or apparantly value free) systems don't teach you this.

Hi, all. Charles, I'll respectfully pose a counterpoint here... Scott's expressed purpose with this blog is to focus on the game industry. Lots of dev studios go out of business, and one solution to that is for game makers to be better at business.

There are tons of great ideas for games out there. There are far fewer focused game implementations that make the creators enough money to pay for their office space.

Charles,
I think Santa is going to put an Edo glass in your stocking! ;)
I think you made some valid points, Scott.

"Hi, all. Charles, I'll respectfully pose a counterpoint here... Scott's expressed purpose with this blog is to focus on the game industry. Lots of dev studios go out of business, and one solution to that is for game makers to be better at business."

I couldn't agree more, which is why I think Scotts initiative, and the many contributions it attracts, is a good thing. That said, not all the answers can be found in business books, and certainly not in as narrow a selection as he's made so far.

A study or appreciation of non-core material such as culture and history can expand the horizons of the possible and assist the development of the whole person. That, I think, is something the modern mindset is losing.

The issues I'm raising are perhaps better raised when dealing with company culture or staff development, which Scott has indicated he may be raising at a later point.

"There are tons of great ideas for games out there. There are far fewer focused game implementations that make the creators enough money to pay for their office space."

Sure. And there are plenty of games that with a degree change of emphasis at the input end of the process that could've been improved beyond measure for a relatively trivial amount of effort. While my comment on Edo glass was a little obscure it was meant it to indirectly highlight the need to look beyond narrow processess and develop a greater depth and breadth of emotional and intellectual wisdom. Again, it's a what versus why thing.

Another great business book is 'Good to Great' by Jim Collins. It's an very well researched book that looks at what differentiates a number of extrememly successful companies from more mediocre ones.

Thanks Scott, all of this is great info. I have noted down all these titles as well as your comments on them. I look forward to more reading material. Currently I am reading 'Secrets of the Game Business' by Dominic Laramee, so far it's pretty informative.

As far as what Charles is saying about understanding society and history is very important in making a hit. As society is a revolving door and spits out rehashes of old 'stuff' done 'new and improved'. This isn't a bad thing at all times, others it is. I think this is where the understanding of society comes in to improve on history and to make something fresh by adding to something old. Am I making sense here?

Although with that being said, I think it is also important to know business as it is unfortunately the most important part of successful company, no matter what the industry. It's all about learning it and applying it to your own need. The more you know, the more you understand, the more you understand, the more you are able to connect, the more you are able to connect, the better you are able to persuade, if u get my drift.

Going back to the door question...

When you start there are three closed doors and you have a one in three chance of picking the big prize and a two in three chance of picking either the big prize or the medium prize. Now, I've never seen this show, but presumably the host will open one of the doors you didn't pick (and uncover the TV) if you picked either the booby prize or the big prize. Otherwise it would be too predictable and everyone would know which prize was behind the door they picked based on what was behind the door Monty opened.

So you now have two doors left to pick from. One has the car behind it, one the donkey. Going purely on statistics, you've got a 50-50 chance of getting the big prize at this point, so it doesn't make any difference whether you switch or stick.

Markus' explanation might make sense at first sight, but it doesn't hold up when you stop to think about it. Yes, there is a two in three chance of the big prize NOT being behind the door you picked at the start. But if Monty opens one of the other doors he's eliminating one of the three choices. He's changing the rules. You now have two doors to pick from, not three, and you know that the big prize is behind one of the remaining doors. So you have a one in two chance of picking the right door.

Anything else is just Monty Hall messing with your mind. ;)

Gestalt, switching is ALWAYS the right move, based on odds, as your odds go from 33.3% to 66.6%. BTW, if you stick with the original door, your odds do not change when Monty opens one of the other two doors -- the odds remains 33.3%. Counter-intuitive, no? ;-)

Regarding the book, Good to Great, I agree it's an exceptional book. Truly one of the best business books ever, but a little more geared for larger companies so I didn't have it on my list.

Geoff, regarding Secrets of the Game Business, I don't rate this book as great, though I'd recommended it for those just getting started in the industry, but definitely not for pros. A few game design books that I *do* recommend include Game Architecture and Design, Game Design Foundations, Chris Crawford on Game Design, and Rules of Play.

"While my comment on Edo glass was a little obscure it was meant it to indirectly highlight the need to look beyond narrow processess and develop a greater depth and breadth of emotional and intellectual wisdom. Again, it's a what versus why thing."

Sorry, we're talking about what sells here, not what's good. Please move along.

"switching is ALWAYS the right move, based on odds, as your odds go from 33.3% to 66.6%"

There's initially a 33.3% chance that it's behind each of the doors. But once a door has been opened and we've seen the TV is behind it, we know that the big prize has to be behind one of the two remaining doors. There's no logical reason why the door we didn't pick should suddenly have twice as high a chance of having the big prize behind it simply because we didn't choose it. If you're still not convinced, try and figure out what happens to the probabilities if you have TWO people choosing doors and Monty opens the one that neither of them picked. :)

That kind of thing works in quantum mechanics, where the act of observing the system can change its properties, but not in quiz shows. Unless some guy is running around back stage switching the prizes around while Monty's blabbering on. ;)

"That said, not all the answers can be found in business books, and certainly not in as narrow a selection as he's made so far."

That's more than fair.

"The issues I'm raising are perhaps better raised when dealing with company culture or staff development, which Scott has indicated he may be raising at a later point."

Ok, I understand where you're coming from now. I think that we're both headed to the same goal, but I'm coming at it from the other side. :)

Switching only works if the host always opens a door. If he opens a door less frequently than two times out of three, stay may be better. In the 0.67..0.99 range switch may still be better but now the strategy of the host when the contestant has picked the great prize matters; does he open the TV set or the donkey more often?

-- "does he open the TV set or the donkey more often?"

If Monty reveals the donkey, then switching is practically a guaranteed win, because it means you've selected the TV and he's definitely not going to reveal the car.

If Monty reveals the TV set, it means your original choice was either the car or the donkey. BUT, your odds still DOUBLE by switching doors.

Gestalt, don't feel bad about not being about to figure this one out -- there have been math professors who have argued endlessly that switching does not matter, before weeks or months later finally figuring it out. This puzzle truly does work against common sense, until you have that "aha!" moment.

Care to take a stab at the other counter-intuitive puzzle I mentioned, which may be even harder to grasp than the Monty Hall one: A couple you've just met has two kids, and they happen to mention that one of them is a girl and plays soccer. What are the odds that their other kid is also a girl?

"your odds still DOUBLE by switching doors"

They don't, and I think I can prove it very simply. :)

For argument's sake, let's assume door 1 has the car behind it, door 2 has the TV, and door 3 has the donkey. Monty knows this, the contestant doesn't.

Let's say the contestant picks door 1. Monty then opens door 2. Ah ha! The TV is behind door number 2. So do we now stick with door number 1 or switch to door number 3? By my logic, both doors are equally likely to have the big prize behind them. But you say that there's now a 66% chance the prize is behind door number 3 simply because that's the one we didn't pick first.

But what happens if the contestant had picked door 3? Monty would still open door 2 and we would still see a TV behind it. I say there's still a 50-50 chance of the car being behind each of the two remaining doors. But you say that there's a 66% chance the prize is behind door number 1 now.

In both cases the setup of the prizes is the same, and the rules of the show are the same. All that has changed is which door we originally picked. And yet you're saying that somehow the chance of the car being behind a particular door is different in these two (otherwise identical) cases, simply because of which door we picked first.

So, are you still convinced Marilyn wasn't pulling your leg? :)

Let's try another one. What happens if we have the same setup as before but now there are two contestants, and each gets to pick a door. Let's say contestant 1 picks door 1, and contestant 2 picks door 3. Again, Monty Hall opens door 2 to show there's a TV behind it. Big "ooh" from the audience. We now know that one contestant has got a car and the other has a donkey. But which contestant has which "prize"? We don't know. Because there's an equal chance of the car being behind each of the remaining doors.

Simply choosing one door over another has no effect on the probability of the prize being behind it. And when Monty opens a door, that doesn't make the door you picked any more or less likely to have the car behind it than the other unopened one.

If the show is rigged (ie, Monty is more likely to show you the TV if you picked the donkey than if you picked the car) then yes, switching doors is obviously more likely to give you the car. But if he's equally likely to show you the TV whether you picked the car or the donkey, then it has no effect on the relative probabilities of each of the remaining doors being the right one.

Unless of course you can prove otherwise. :)

Gestalt, what you're missing is that when Monty opens one of the doors, it's odds are transferred to the other door that you did not pick. This is because Monty knows not to pick the door with the big prize.

You can build a computer simulation of this that easily shows that switching is always the right move.

Anyway, I'll not push this any further. But, I'm sure you can look it up on the net somewhere, as this has become quite the famous puzzle and has tripped up a lot of bright people.

Scott, I like this place. :D

Okay Gestalt, Door One has the car, Two has the TV, and three has the Donkey.

You pick Door One(car), he opens Two(TV). You switch to Three(donkey) and lose. A loss.

You pick Door Two(TV), he opens Three(donkey). You switch to One(car) and win. A win.

You pick Door Three(donkey), he opens Two(TV). You switch to One(car) and win. A win.

It totally goes against common sense, but it's right. Switching does give you a 67% chance of winning. To prove it to yourself try using Scott's suggested coin method. Or use two pennies and a Quarter. After all, you're not going to 'lose' anything, so why worry about aiming for the TV? Go for gold.

Ah, the wonders of Google reveal that the puzzle Marilyn explained had one door with a car behind it and two with goats. Yours has a car, a goat and a half-car, half-goat. ;)

Once you start adding TVs into the mix it's harder to figure out what's going on. But the two goats and a car explanation does make sense when you read through it all point by point, and yes, I was wrong. :)

Does your version of the puzzle work? It *might* do. But the information Monty is giving in your example is slightly different to how it works in the classic version of the puzzle, and this information is the key to how the puzzle works.

If you pick the TV, then Monty either has to do nothing or show you the donkey. So if he shows you the TV, you now know that you have either a donkey or a car behind the one you have chosen. Whereas in the classic version, whichever door you pick Monty can always show you a donkey behind one of the two remaining doors.

Unfortunately some of us have a deadline to get our game finished by, so I don't have time to work through all the possible combinations of car, donkey, TV and player choice of door right now. Any volunteers? :)

Looks like Jeffool beat me to it. Thanks. :)

I would say the odds are "poor" that the second child plays soccer, simply because if both were girls, a person would tend say "one of our girls plays soccer", instead of using the singular "girl".

If you’re asking for specific numerical odds, well that's beyond the reaches of my intelligence ;)

Oh for an edit feature. I guess I was thinking too hard. What I meant to say was

"I would say the odds are "poor" that the second child is a girl"

Gestalt, fundamentally, there's no difference between two goats, two TVs, or a TV and a goat. The key is that the contestant always wants to win the car. Switching gives you the best odds to do so.

Dan, you could say the odds are "poor" and be fairly accurate. But, what exactly is the percentage. Hint, it's one of these choices: 10%, 20%, 25%, 33%, 50%, 67%, 75%, 80%, or 90%.

Here's the exact same puzzle, stated differently so that we don't get hung up on the gender tendencies of playing soccer: You're blindfolded. A friend throws two pennies in the air and they fall to the ground, both lying flat. Your friend tells you "One of the coins is heads-up." What are the odds that the *other* coin is heads-up?

"The key is that the contestant always wants to win the car. Switching gives you the best odds to do so."

Yeah, that's the bit that had me confused. Once you realise the TV is just a cunningly disguised goat, it makes more sense. ;)

Wow. On the children or coin-toss question I'll admit to having no clue, but I'll toss a random guess of 75% (because it's a pleasing number) that the other child is a boy, and that the 'other coin' is NOT heads up.

Ahh that makes more sense now.... everything is clearer when you view it in cards or coins.

I've herd the door problem before, but with the context of a dealer dealing one ace and two queen’s face down, and the individual making the choice is trying to pick the Ace.

With regards to the child/coin problem, in such a system it would seem that there is a 25% chance that the second child is a girl, or the second coin is heads.

My reasoning is that with two coin flips there are 4 possible arrangements of the coins. The chance of having them both be girls, or both is heads is 1 in 4. I think the logical trick in this one, is that you are not calculating the chance that they are both the same, but that they are both a specific type, aka. Heads or Girls.

Charles E. Hardwidge wrote:

"Winning and losing are both opportunties even if the norm of typical thinking likes to condition us differently. That's what bothers me about Scotts reading list. It knows the cost of everything and the value of nothing."

It knows the value of success, and painful cost of trial & error. If you want books which better romanticize the human condition, try philosophy or a similar subject with a knack for emotionally intellectual fluff:

Dream of Reason

Basic Writings of Nietzsche

Hero With A Thousand Faces

Keep in mind Scott isn't intentionally excluding certain kinds of books, he just hasn't given his entire list yet. He's starting with business because that's a subject in which most people seem to have a significant deficiency (more like abject ignorance). I'm sure he'll be posting links on story and character creation at some point in the future.

If you're not finding your favorite subjects or books listed here which you feel are relevant to game design or the industry, feel free to list them.

"A friend throws two pennies in the air and they fall to the ground, both lying flat. Your friend tells you "One of the coins is heads-up." What are the odds that the *other* coin is heads-up?"

Ooh, I know, I know! This is like a simpler version of the Monty Hall puzzle.

At first glance you might say that the other coin has a 50% chance of being heads (or tails). After all, it doesn't matter what the first coin does, this second coin just lands on heads or tails with equal probability, independent of the first one.

However, as with the Monty Hall puzzle, when you look at the probability tree everything becomes a lot clearer. I'll just flatten the tree to the 4 possible combinations, all happening with the same probability:

Tails & Tails

So, since we know one of the coins is heads-up, there is only a 1/3 chance that the other coin is heads up too (as the other two combinations with H in them are HT and TH).

Hmm, so, you can't look at the components of the system independently, you have to look at the system as a whole.

Great book list by the way. I have about half the books on Scott's book list (on the 3D Realms site). 'Good to Great' and 'Positioning' are among my favourites. Another one I'm always recommending is 'Influence - The Psychology of Persuasion' by Robert Cialdini ( http://www.amazon.com/exec/obidos/tg/detail/-/0688128165/ ). Scott would like this one I think...

I think the problem that most people have with probability is not realizing that it is heavily dependent on the given information, and not just the absolute realty of the situation.

a trivial example:
I flip a coin. What's the chance it's heads?
I flip a coin again, but show you that it landed as tails. What's the chance that it's heads?

"A friend throws two pennies in the air and they fall to the ground, both lying flat. Your friend tells you "One of the coins is heads-up." What are the odds that the *other* coin is heads-up?"

Not enough information, really. If your friend is always telling you the number of coins that are heads-up, 0%. If your friend will always tell you one of the coins is heads-up when that is true, 33%, as per the last poster's answer. But the most reasonable assumption is that your friend is randomly picking a coin to tell you, and that the odds of the other coin being heads-up are 50%.

Aaargh.....!
Is the other kid a boy or a girl?

I've been busting my nut over this one and I've got a feeling it is 1/2 BUT...

Initially it would seem to be a 50/50 chance boy or girl, but the mention of the girl playing soccer makes me wonder if it is just a red herring.
If you have two kids and the chance is one of them will be a girl and one will be a boy, then one of them IS a girl, does that mean statistically it is going to be a boy? Or does the fact that the coin is effectively flipped once again to decide the next kid mean it's still 1/2?

Good brain teaser though :)
Interesting selection of books too, most people would probably have been expecting game design/programming books instead of marketing/business books.
(Should I throw in the token "When is Duke coming out?" question or not? Now that I've already asked it, what are the chances of me asking again?)

"As far as what Charles is saying about understanding society and history is very important in making a hit. As society is a revolving door and spits out rehashes of old 'stuff' done 'new and improved'. This isn't a bad thing at all times, others it is."

Geoff, that was the other half of the reason for taking a more rounded approach to reading material. It helps give what you're reading some sense of perspective and context. As I said earlier, maybe I'm just being difficult. Personally speaking, I'm never happy with the promotion of value free systems or methods without people being made aware of the consequences.

"Keep in mind Scott isn't intentionally excluding certain kinds of books, he just hasn't given his entire list yet. He's starting with business because that's a subject in which most people seem to have a significant deficiency (more like abject ignorance)."

That's not something I'm disputing, Eric. For the record, Scotts recommendation of "Patton on Leadership" reminds me that I'd made a note to grab a book on Patton at some point.

"If you're not finding your favorite subjects or books listed here which you feel are relevant to game design or the industry, feel free to list them."

I have some suggestions that might be worth making at some point, if I can get over the hurdle of writing up the abstracts. While some need no introduction others might need a little explaining. (I've just checked. One of my top recommendations went out of print years ago. Damn.)

The answer to the two pennies puzzle is exactly as Luc said. The key is that you're not told which coin is heads, only that one of the coins is heads. Individually, each coin has a 50/50 shot of landing heads-up. But the puzzle forces you to consider both coins as a pair, and the breakdown is HH, HT, TH and TT. If one coin in the group is H, then the TT possibility can be thrown out the window, leaving three possible outcomes. Only one of those three outcomes has the other coin heads-up, thus the answer is 33.3%.

One final puzzle with a counter-intuitive answer: One of Carmack's super charged Ferrari's cruises rather slowly for one lap around a one-mile long oval track at 30 miles/hour. How fast would it need to go on its second full lap to average, between both laps, 60 mph?

This is the coolest blog, where else can you do cool logic problems in the comments? This is great preparation for all those would be Microsoft employees by the way :)

I'm 0 for 2 on these things, but the exercise is enjoyable enough that I’m willing to risk publicly exposing thinking ability (or lack there of).

I’m going to take a really “left field” approach to this problem, since the other two have defied standard approaches.

Fist off, 60mph is one mile a minute. That means that Carmack should finish the 2 mile course in two minutes in order to average 60mph for the trip.

However Carmack traveled the first lap at 30mph. Traveling at 30mph Carmack is traveling at a 1/2 mile a minute. But he travels a full mile on the first lap, consuming a full 2 minutes. So in essence, for Carmack to average 60mph for the full 2 lap distance after doing one lap at 30mph, he would have to complete the 2nd lap instantaneously?

"This is the coolest blog, where else can you do cool logic problems in the comments?"

Well, there you go. Getting first place in the mind puzzle market is clearly more sellable than discussing the relative merits of a reading list. It also proves the merits of being more widely read. Everyone's a winner.

The 2nd problem is actually a fair standard probability problem of p(E|F) or probability of E given F.

Which is calculated as p(E and F both happening) / p(F)

or the probability of both happening divided by the probability of the given event.

Lets say that E is both being girls and F is one child being a girl.

Looking at the possible outcomes (B-B, G-G, B-G, G-B) we see that there is a 3/4 chance of there being at least 1 girl, there is also a 1/4 chance of there being two girls.

so the probability of E given F is 1/4 / 3/4 or 33%.

The trick to this problem is that logically you assume that order of children doesn't matter. Thus you may think that G-B is the same as B-G. While both outcomes would result in the same outcome (the parents saying they have a girl and having the other be a boy) the fact that both instances occur is a factor. Instead of an even distribution you have B-B occuring 25% of the time, 1 boy 1 girl 50% of the time and G-G 25% of the time. Since the distribution is uneven it throws the answer of from the intuitive 50% answer.

For the race problem I'm going to have to go with 1920.

We know that he has gone 30mph for 2 minutes. To average 60mph he needs to go 90mph for 2 minutes. However if he goes 90mph he has gone into a 3rd lap. So he must go faster... much faster.

Doing some average calculations we can find that if he goes 30mph for 2 minutes and 1920mph for the 1/32 of a minute it takes to complete the lap (damn that's fast) then he has gone 30mph for 64/32nds of a minute and 1920 for 1/32nd of a minute.

So lets compute the average (30*64 + 1920*1) / 65 = 60.

How did I get 1920 and 1/32 of a minute to try out the problem? good old base 2. I started at computing the avg mph for 30 for 2 minutes and 120 for 1/2 a minute (2 laps) and kept doubling the 2nd lap speed (240, 480, 960, and finally 1920) which halved the lap time (1/2, 1/4, 1/8, 1/16, and 1/32).

Not the best explaination but I think it's right... and carmack's ferrari is pretty nice if it can handle those speeds.

"But the puzzle forces you to consider both coins as a pair, and the breakdown is HH, HT, TH and TT. If one coin in the group is H, then the TT possibility can be thrown out the window, leaving three possible outcomes."

Yes, but the three possible outcomes only have an equal chance of happening if your friend always tells you one coin is heads up when that is true (which is an absurd assumption). If he randomly picks a coin to tell you, the three possible outcomes do not have an equal chance of happening - specifically, HH will show up twice as often as the others, because half the time when you get HT or TH he will tell you one of the coins is tails side up.

1/4 of the time you get HH, he tells you one is heads.
1/8 of the time you get HT, he tells you one is heads.
1/8 of the time you get HT, he tells you one is tails.
1/8 of the time you get TH, he tells you one is heads.
1/8 of the time you get TH, he tells you one is tails.
1/4 of the time you get TT, he tells you one is tails.

Half the time he'll tell you one is tails, which he didn't this time so we can ignore those possiblities. That means half the time it was HH, one quarter of the time it was HT, and one quarter of the time it was TH, so the other coin will be heads half the time.

Charles E. Hardwidge, you are trying *way* too hard with the pseudo-intellectual righteous indignation thing.

Thanks for sorting out that boy/girl one. It makes sense (in an odd sort of way).

If I had a ferrari, I don't think I'd care how fast I had to go :)

"you are trying *way* too hard with the pseudo-intellectual righteous indignation thing."

That's an easy judgement to make if you don't have a complete grasp of the picture, and for that I'll accept the responsibility by my perhaps not explaining as clearly as I might. Let's take Scotts suggestion of "Patton on Leadership," as an example.

From the brief exposure I have, many of the things Patton said are bang on. Taking them as verbatim isn't always advisable. The man himself wasn't just a walking rulebook, he had a depth of experience and values to back them up. That's where the overwhelming majority of box tickers fall flat. They have knowledge without understanding.

Even though practice makes perfect it still won't tell you why something should be done. Issues of social and cultural responsibility aren't seemingly a high priority for most business today which is why companies like Disney and Coca-Cola operate as they do.

Quote:

Okay Gestalt, Door One has the car, Two has the TV, and three has the Donkey.

You pick Door One(car), he opens Two(TV). You switch to Three(donkey) and lose. A loss.

You pick Door Two(TV), he opens Three(donkey). You switch to One(car) and win. A win.

You pick Door Three(donkey), he opens Two(TV). You switch to One(car) and win. A win.

End quote

Umm, is the assumption stated as this Monty fellow "never opens the Car door for you"? In that case, you're missing the case where:

You pick Door One(car), he opens Three(donkey). You switch to Two(TV) and lose. A loss.

So we're back to switching looses half the time. Are you sure the rule isn't that Monty "never opens the car, and always favours the TV?"

The way I see it, if the rule is only that Monty _never_ discards the big prize, then practically your first choice is completely irrelevant. Probability of successful first choice is 100%. The second choice is always car / not car.

This is still illustrated by the 1000 doors case. If he slams open all but your choice and another, his criteria was "open all doors bar 2: the car, and another". The cases are either that

you were on the car, and he randomly selected another door.
or that you were on an irrelevant door, and he selected the car.

The probability at the second decision still seems to always be 50/50 to me.

The three door problem is interesting.

Case1 : I select the car door; Monty shows me the TV.

Case 2: I select the TV door; Monty shows me the goat.

Case 3: I select the goat door; Monty shows me the TV.

There are exactly two situations in which Monty shows me the TV. In one of them I selected the car, in the other I selected the goat. Therefore if Monty shows me the TV, I have a 50:50 chance that I selected the goat and therefore a 50:50 chance that switching will win me the car. No improvement.

All this tells me is that if Monty shows me the goat, I should switch choices.

Speaking more generaly, before I make any choice it is true that in some circumstances switching my choice after monty shows me a duff prize will improve my chances, but in other circumstances it makes no difference. Therefore as a general rule, switching choices will either make no difference to my chances of winning the car or will improve my chances. Therefore, [B]before I start playing the game[/B] adopting a strategy of switching choices when given the chance, regardless of what Monty shows me, leads to a marginal improvement in my overall odds. However it only actualy improves my odds if Monty shows me a goat.

Simon Hibbs

One way to look at it is that the fact one of the coins is a head has already been determined. Working out what the odds would have been before that information was given out, or what the decision tree was from before the coins were tossed, is irrelevent and will distort any analysis of the situation. In other words, the fact that one of the coins is a head is a pre-determined fact, not a random event. Therefore it can have no effect on the probability of the other coin being a head too. Therefore that probability is 50:50.

Simon Hibbs

"You pick Door One(car), he opens Three(donkey). You switch to Two(TV) and lose. A loss"

That's true he did leave out some cases here are all the cases:

1)Pick Car: shown donkey: stay: win
2)Pick Car: shown donkey: switch: loss
3)Pick Car: shown tv: stay: win
4)Pick Car: shown tv: switch: lose
5)Pick Donkey: shown tv: swtich: win
6)Pick Donkey: shown tv: stay: lose
7)Pick TV: Shown donkey: stay: lose
8)Pick TV: Shown donkey: switch: win

From here it looks to be 50/50 but lets look at the distribution of probability.

You may think each event is equally likely correct? That's not true. The first 4 events are as likely as 5 and 6 or 7 and 8. This is because the probability of the event relies on what you picked first. You obviously have a 1/3 chance of picking the car, a 1/3rd chance of picking the tv and a 1/3rd chance of picking the donkey.

So since there are more possible outcomes when picking the car the probability of each of 1-4 are less (1/3*1/4 or 1/12) while 5-8 each have a 1/6th possibility.

So if we create an even distribution (by counting 5-8 twice (for 2/12 of the time) of the events we result in

1) Car: Donkey: Stay: win
2) Car: TV: Stay: win
3) TV: Donkey: Stay: lose
4) TV: Donkey: Stay: lose
5) Donkey: TV: Stay: lose
6) Donkey: TV: Stay lose

7) Car: Donkey: Switch: lose
8) Car: TV: Switch: lose
9) Donkey: TV: Switch: Win
10) Donkey: TV: Switch: Win
11) TV: Donkey: Switch: Win
12) TV: Donkey: Switch: Win

Here we can now clearly see the winning 2/3rds of the time by switching.

The problem you have with this problem is you assume all outcomes are equally likely. However you have to understand that outcomes that involve picking the Donkey or TV first are going to be twice as likely because once you pick the TV or Donkey there is only 1 other door he can reveal instead of 2. When you pick the car he has 2 doors he can possible reveal (which we assume are at random) so there are twice as many outcomes but that doesn't affect the probability you picked the car first.

The (first and 2nd) are both problems of distribution. Realistically since it's pretty much an a,b,or c choice you assume that the distributions are equal across all choices. However you fail to realize that some outcomes are more likely (for example choosing the donkey and seeing the tv is more likely than choosing the car and seeing the tv).

Moving on to the coin problem again the trick to this is that it is not said 'the first toss is heads what's the 2nd toss' it's simply put that one of the 2 is heads. If the question was what's the probability that you flip 100 coins and none of them are tails then you'd certainly not say 50%. The same applies here. What's the probability that you flip two coins and neither of them are tails? It's certainly not 50%. Now you are given that one is heads. That immeditally rules out T-T occuring. So we are left with H-T, T-H and H-H occuring. That means our H-H situation will only occur 1/3 of the time instead of 1/2 or 1/4 (probability if you weren't given any info except for 2 flipped coins).

So this problem is a matter not as much of distribution as it is possible outcomes and grouping. While each coin still has 50% chance of being H or T together the chance both of them being H is different because you were not given order and you can rule out one possible outcome.

Really great puzzles Scott (it's fun to see people worm with the intuitive answer) and great site.

Anyone have any other guesses/reasons for problem #3?

---- "does he open the TV set or the donkey more often?"

-- "If Monty reveals the donkey, then switching is practically a guaranteed win, because it means you've selected the TV and he's definitely not going to reveal the car."

No. It means you selected the TV set or the car.

-- "If Monty reveals the TV set, it means your original choice was either the car or the donkey. BUT, your odds still DOUBLE by switching doors."

Again, no. Your odds only double if Monty is _absolutely required by the rules_ to always open a door. If he is allowed to decide, the game changes. What if he only revealed the TV set because he knew you picked the car?

-- "A couple has two kids, and they happen to mention that one of them is a girl and plays soccer. What are the odds that their other kid is also a girl?"

This one is similarly deep. The odds depend on the "strategy" that was used to come up with the "one a girl, plays soccer" statement.

* pick a particular kid. report gender. 50% (soccer is irrelevant.)

* pick a kid at random. report gender. pick a characteristic that this kid has ("plays soccer") - in other words, this gives no information. 33%.

* pick a kid at random, report gender, report whether he/she plays soccer. No simple answer, calculation depends on many factors.

You are reading too much into the problem. Assume the simple answer because most of the time that's what it's going to be. Monty is always going to open a door. Assume at random, there's no indication otherwise. Even if it's not random, to the contestant that doesn't know better it is so he/she is still playing the odds by switching. The parents simply picked a kid at random (there was no indication of it being the 1st or 2nd child) and soccer is an irrelevent addition to the problem (soccer has nothing to do with 2nd child's gender) so there's no reason to assume anything other than 33%.

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