Let me reiterate what I have already said. If you consider that this game is one of a series of games (i.e. this game is played daily), then the outcome is different then if you consider that the game only happens once. If you consider this scenario as one of a series then it becomes a question of economics more than purely math. Economically you should assume that the show wants to maximize profits and therefore minimize costs. If it is possible for the chance of receiving a car to be 1/2, then the show will make the probability 1/2.
As I stated before it is possible to make the probability of winning by switching to be 1/2 if Monty sometimes opens up all the doors without giving the contestant a chance to switch. And if you've watched "Let's Make a Deal" you'll see that he does in fact do this. Sometimes the contestant is given a chance to switch and sometimes he reveals what they get after just one pick. The show very much has an air of unpredictability. You don't really know what Monty is going to do next.
The only real way to know if the producers are playing some metagame or not is to actually observe how often the game is played in this fashion and record if the observed probability is closer to 2/3 or 1/2 (or perhaps it is close to neither). If the observed probability is close to 2/3 then we can assume there is no metagame, otherwise we must assume that the producers are somehow affecting the probabilities of the game.
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Thread: Simple puzzles to stump people

09212007, 05:16 AM #141My wife and I made a game to teach kids about nutrition. Please try our game and vote for us to win. (Voting period: July 14  August 14)
http://www.revoltingvegetables.com

09212007, 05:36 AM #142
Tell that to those who have been wrongfully convicted in criminal trials on "common sense", i.e. various forms of the base rate fallacy. (Link)

09212007, 05:38 AM #143
I agree with different winning strategies for onetime and repeated games. I guess I confused what you ment to be the strategy for the onetime vs. repeated game. We would have to know what effect different outcomes would have on the audience, i.e. would the popularity decline or improve depending of certain winning ratioes, etc. We can either try to model themselves or accept that the producers go by their own model.
I perhaps wrongly assumed that this is a cooperative game, when it's actually a zerosum game. It's only cooperative outside the stated game mechanics, i.e. how favourable an impression the host makes vs how much money it costs to run the show.
Some nonzero sum games have proven best outcomes for cooperative play when it's uncertain (in a certain probability range) whether there will be a new round. The stability of such solutions may or may not fullfill various gametheoretic criteria.
Interestingly, if a round is known to be a last round, with some game types and payoff matrixes, the winning strategy is to go for personal profit.
I haven't watched the show and this fact didn't come up in my searches. This does put a different spin on the game. I have taken the description of "Monty Hall Problem" as an accurate description of the game and I feel like a fool now :P
True.

09212007, 10:54 AM #144
While not a replacement at all for what Ygolo posted, here's the logic behind the "children" questions.
The tree for the 3/4 problem (at least one girl, at least one boy, but can be used for similar problems);
This is how we think of the problem. However, it's flawed when conditions are added (here is the 2/3 problem
The red line denotes what isn't possible given the rules of the problem. That makes only three possibilities that are allowed, with two being accurate.
The way the mind tends to think of the tree, however, is like this;
Which is why we instinctively think of it as 1/2. However, the assumption of the boy first is incorrect.
Here's another way of looking at it, if this doesn't make sense. Lets say that you pick 1000 couples that have two kids. Of that, there should be an even split of 4 (BB, BG, GB, GG). But then you say  well, I know this couple that has a boy. So you go from;
to
In other words, you have changed your sample group.
(Forgive the drawings, I only have excel to work with here )Last edited by ptgatsby; 01212008 at 12:14 PM.

09212007, 11:28 AM #145
What I don't understand are why BG and GB are considered separate possibilities. They are both one boy and one girl, just the order is different.

09212007, 11:37 AM #146
Think of it as this tree;
The x at the top is "Sex" between the couple  from this can either come a boy, or a girl. Up to here it makes sense that there are two different outcomes.
Now, inside each outcome, repeat sex  another B or G (another fork). That makes the BG the result of one outcome and GB the result of another. Both can be viewed as "one boy and one girl", but both are not "a boy, then a girl". When calculating how many possible outcomes, you have to measure all of the different combinations something can happen (a boy, then a girl + a girl, then a boy = a boy and girl).
Does that make sense?

09212007, 11:38 AM #147
There are lots of choices. If you choose BG and GB to be the same, you won't have a uniform probability distribution. You're derivation would go a little different.
Keep in mind that you are deriving this distribution based on what is given in the problem (The probability of having a boy being 50% and a girl 50%. Hermaphrodites and others are 0%, if you were to consider them)
Accept the past. Live for the present. Look forward to the future.
Robot Fusion
"As our island of knowledge grows, so does the shore of our ignorance." John Wheeler
"[A] scientist looking at nonscientific problems is just as dumb as the next guy." Richard Feynman
"[P]etabytes of [] data is not the same thing as understanding emergent mechanisms and structures." Jim Crutchfield

09212007, 12:01 PM #148

09212007, 12:18 PM #149
Well... that is a real life situation, really Forget all about the problems, this is the literal representation of what could happen. Like ygolo says, this is about distribution.
Tell me this:
What is the chance of having a boy and a girl in a family? Is it the same as having two girls or two boys?

09212007, 12:32 PM #150
I am not sure what the real life probabilities are for having a boy or a girl, but if it is 50%/50% roughly and if the gender of the first an second child are truly independent (part of my experience tells me the some family are prone to girls and other to boys, but without looking at demographics it's just a blind guess), then it will come out that way in real life.
But you can "simulate" the problem as follows. Find a "fair" coin (flip the coin a bunch of times to see if the odds are 50%/50%). Now let tails represent a boy, heads represent a girl.
The experiment is as follows: Flip the coin twice, record the outcomes. The possible outcomes are HH, HT, TH, TT. (Or if you want you could map the outcomes to "both heads", "both tails", or "on head one tail".)
Repeat the experiment many many times (say a 100 times). Compute the percent of times you get HH, HT, TH, and TT each separately. You should get that the percent of times that you get each of the outcomes is about 25%. If you repeat the experiment a 1 thousand times, you should be even closer to 25% each.
EDIT: I forgot you knew how to code. You could probably run a long computer simulation using a random number generator if you wanted.
Accept the past. Live for the present. Look forward to the future.
Robot Fusion
"As our island of knowledge grows, so does the shore of our ignorance." John Wheeler
"[A] scientist looking at nonscientific problems is just as dumb as the next guy." Richard Feynman
"[P]etabytes of [] data is not the same thing as understanding emergent mechanisms and structures." Jim Crutchfield
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