Probability question...

[edit: wait a second. I got something wrong there, didn't I? :-))]

I didn't want to call the subject trivial because it might be insulting to less mathematically members.

Anyway, I kind of enjoyed the chance to exercise my math skills. I don't get to very often. I even took it further and answered "If you roll Nd4, what are the odds of getting at least 2 4's" just for fun.

Ok I will agree that it is not "trivial" and I used trivial with respect to my years of taking Math for my BS. And that fact that I have not used "math" skills in a long time, proves through the various posts that the topic is not trivial (though about 10 years ago, it would be trivial for me if my math was fresh in my mind).

Having seprate degrees in Math and CS doesn't make sense to me. How did you cover numerical analysis - splines, machine epsilon, DCTs, etc? Just going over my 3rd and 4th year courses, they're all mostly math.

Out of curiosity, what do you think of www.itallconnects.com ? Since you're a programmer, what sort of algorithm would you use to find the paths through several connections between items?

Why does having two seperate degress not make sense? I wanted to learn higher level mathematics, like abstract algebra and theory of geometry, advanced calc, linear algebra(which deals with numerical analysis), set theory and logic (and the like) so I took additional math classes that would let me obtain my BS in math. keep in mind that just having a cs degree, does not cover all the useful mathematics that one could use in development. So I decided to take higher mathematics in the hopes to use it for graphics(and similar developement), but I am currently working on driver applications, which use basic algorithms (or even worse have a predefined framework as prodivided by MS, Apple, etc, thus no special math skills needed).

Being the odd programmer I am, I might start with Djikstra's algorithm (or a variation) for what is being done on www.itallconnects.com. Though I will say there are other options and since we did not state a requirement as to how the data is store, my choice is purely based on finding the shorest(quickest) path/connect from source A to B. Depending on the method of storage, there is most likely an optimization or alternate algorithm that might fit better for finding the connections.

I have a feeling this thread would have been even longer if the probability question was "Let's Make a Deal," a situation where all intuition tells you the correct answer is wrong, to the point they were able to have it on a game show.

It is trivial for me now. I just meant there are people who consider 2+2 non trivial.

As far as your skills getting stale, do you still remember how to do related rates in Calculus? That's the first thing I noticed I forgot, which scared me quite a bit. I bought a Schaum's outline of Calculus pretty quickly and refreshed myself on it.

I wasn't saying it doesn't make sense for you to have taken the two degrees, I mean I don't understand how the school can separate CS from math to offer a separate CS degree.

Please don't take offense at my questions. It's very clear that your math and computer skills are quite strong. It's just that the way CS was taught at my school you can't have CS without having the higher mathematics, so I want to understand how you can. On some level you seem to agree with me since you did take both degrees.

You said you studied numerical analysis as a part of Math. How about formal languages; that's more clearly CS and just as heavy on math. How can you cover P=NP formally without heavy doses of set theory?

The site does use Dijkstra's algorithm, which is a part of Graph Theory, not CS. If you had taken CS without Math, would you have been able to answer that (I'm trying to see where the line is drawn between the two)?

Right in one. On the forum where this problem was originally raised, the discussion went for 5 pages, I had to write a program to show it worked empirically over 10,000 runs, and then I had to go so far as post the code, heavily commented, just to prove I did it right. Fun times.

It's an interesting choice for players, though I'm not sure how feasibly it could be integrated into a board game. Cons are analysis paralysis and a feeling of "solving the game." These inherent problems (which work to an advantage in a game show) might out weigh any advantage it has (namely, it's an interesting mind-twist).

However, if probability-related choices like this were to arise dynamically, where each situation seemed markedly different from the last (in other words, if only the core concept behind this problem was implemented, and not the game choice itself), and possibly if an element of time was overlaid to limit AP, it could work. Possibly.

The problem has been around for far longer than the formu where it was "originally raised" has existed.

Also, that code sure wouldn't be enough to convince me. It's only pseudo-random, how do you prove its not the result of a strange quirkin your random number generator?

How about this example. I have 3 playing cards, the ace of spades and the red kings. I shuffle them, and you want to pick the ace. You pick one at random without looking at it. I look at the other two, pick one that's not the ace and show it to you. Then I offer you the chance to trade yours for the one I didn't show you. Do you trade? This is exactly the same as the quesiton given.

Now let's modify it a bit. We use the entire deck. You're still trying to get the Ace of spades. You draw one card without looking at it, I look at the remaining 51 cards and show you 50 that are not the ace of spade. I now give you a chance to trade your card for that 51st card. Do you trade?

How about if there's a million cards in the deck?

Jason

For those interested (probably no-one), many of the raditional questions about probability comes down to Bayes Theory:

Pr(A|B)=Pr(A and B)/Pr(B)

In other words, the probability that A happens, given B happens, is the probability of them both happening, divided by the probability that B happens.

In the case where A and B are independent, the probability of them both happening is the product of their respective individual probabilities, such that

Pr(A|B)=Pr(A and B)/Pr(B)
=Pr(A).Pr(B)/Pr(B)
= Pr(A)

In other words, the probability of A occuring has nothing to do whether B has already occurred.

One example in in tossing a coin.
A= head on second toss
B= head on first toss

Thus , if A and B are independent, we can say that the probability of getting a head on the second toss does not depend on whether the first coin toss was a head or not.

However, if we were trying to predict the probability of a head based on past events because we thought the coin might be biased for example, then it is reasonable to have your estimate of probability based on thoe results of previous coin tosses.

I'll stop there before you all fall asleep ... (oops ... too late!)

Wow... I really want to play devils advocate, just to keep this topic and thread going. But I will withhold my silly desire ;) before I annoy everyone!

Hello

When the problem with the three cards was presented to me (several years ago), I did not get it, so problem was presented like this instead.

You have a normal deck of cards. Pick one randomly without looking at it. Now let someone remove 50 cards (leaving one) from the reaming deck. He can not remove the Ace of Spades. Who would not trade? This is the same thing.

/Johan

Sounds like something that could turn into an interesting discussion, but I'm not entirely clear on what you mean. Could you clarify your statement and perhaps give a few examples of the "holes" you are talking baout?

- René Wiersma

Good example Zzzzz, and an interesting case. I agree with you that in this case it is not really necessary to patch the hole, because it doesn't break the game and even adds some flavour. I can see why it might be cool to store wizards spells in a relic.

On the other hand, I think you must be very careful with how you word this in the rules. One player might argue that his Theurgist is able to store wizard spells in a relic because the rules don't explicitly forbid it, but since this is not intuitive it could easily turn into a heated rules argument with other players arguing the opposite.

In this particular case I think you have to make a very clear distinction between "wizard spells" and "priest spells", and make clear that they both belong to the superclass of "spells". Some things might apply to all spells or only to a particular subclass of spells (i.e. "wizard spells" or "priest spells"). If you use that wording consistently throughout the rules, there isn't really a "bug", just an undocumented feature.

In other words:

Class "priest" can use "priest spell"
Class "priest" can put "spell" into relic
Class "priest" can release "spell" from relic

Class "wizard" can use "wizard spells"

Class "theurgist" can use "priest spell"
Class "theurgist" can use "wizard spell"
Class "theurgist" can put "spell" into relic
Class "theurgist" can release "spell" from relic

- René Wiersma

Yes, that's exactly the sort of thing I was talking about.
Here's one of my examples (although less extreme than that one!)
In All for One, players undertake missions, which basically involve "pick-up-and-deliver" tokens. In an earlier incarnation of the game, players had to commit to doing a particular mission by putting the card down in front of them (face-down, so no-one else knew what it was.) When they reached the destination, they revealed the mission and scored the points.
However, I had deliberately left a hole such that a player could arrive at their destination and only then declare a mission (which they then immediately completed, of course.) One of my playtesters picked up on this, and soon everyone was doing it (I should point out that I didn't choose to do it myself to see if it was picked up.)
Subsequent development has removed that element from the game now (you don't ever declare missions) so the issue doesn't arise any more, which is a somewhat drastic solution to the "problem" ;-)

It seems to me that the only time a true loophole exists is when the designer did not intend on it being there. Any rules the designer purposefully leaves unfixed to allow players flexibility with their strategies isn't a loophole, its an intentional rule, or strategy.

I agree with Seth that if the loophole breaks the game it is bad and I can't see why a designer would purposefully leave one in. The loopholes that concern me as a designer are the ones that I didn't notice until playtesting, at which point someone proves that my rules seem to be very broken in one way or another.

This topic does lead me to an interesting game idea (not one I could develop probably), where one of the points of the game is to discover new strategies that exploit specific weak points in the rules. These weak points would have to be well hidden, and the strategies that could be derived there from would also have to be difficult to discover. I think such a game would be fun because it would let me as a strategist discover ways to use the rules to my advantage (something my AD&D friends don't like me doing in that game). A game like that could be interesting but I can't fathom developing it.

That is true, and I do love a good game of Magic (though I haven't bought cards in a lon time). Would it be possible to create a game that gives as much flexibility but doesn't require the same amount of time, effort and MONEY as Magic? A board game that has flexible enough rules to allow players to delve into different strategies in their exploration of the rules.
Magic is successful at this because there are hundreds of variables in the different cards and they are constantly adding new cards and new rules (which in turn get rid of old cards and old rules). This creates a very flexible system where players can exploit, explore and strategize to their "pocket book's" content. Would the same need to be true for any game to have such flexibility and ever changing strategies?