Oops I broke the laws of math.

[leonid]

Re: Oops I broke the laws of math.

I think you should focus on finishing college apps, instead of making threads >.>


@emerald: How did you get those numbers? Are there any specific patterns in it?

[emerald000]

Re: Oops I broke the laws of math.

Got them by looking at the expected outcome of drawing a card and comparing it with the current amount amassed. If the first is bigger, draw. Otherwise, stop.

There has to be a pattern, but I haven't work to find it.

[bluguerrilla]

Re: Oops I broke the laws of math.

One of my math profs actually asked us to solve S = Sum[1/n^3,1,inf] and told us we'd get a prize if we solved it. The next class he elaborated that the prize would be eternal glory because it's just not gonna happen.

One of the neatest derivations I've ever seen tho is starting with the geometric series and ending up with the solution to S = Sum[1/n^2,1,inf]. You just have to know basic calculus to figure it out.

[Relambrien]

Re: Oops I broke the laws of math.

proof by induction

That problem is well below the difficulty of 15-251, isn't it.

Ugh, next semester is going to suck.

[bLasTamos]

Re: Oops I broke the laws of math.

Wild guess at card game: stop when the probability of drawing a red card times the money left on the deck is lower than 1.

[MrRubix]

Re: Oops I broke the laws of math.

*Goes hide in a corner...*

Let me modify that a bit.

Expected outcome: 2.62447554899

Excellent answer -- 2.6244755 is right, although technically red cards have no impact over the stopping point -- only the black ones (the optimal score to quit at is a non-increasing step function of the number of black cards drawn), as drawing black cards lowers the optimal score to stop.

Basically, the optimal strategy is the same as locating an optimal stock price exercise boundary for an American-style option (aka you have to work backwards through a decision tree while calculating expected payoffs for continuing to play versus stopping at a given point). For example, for a 2-card, 4-card, 6-card, and 8-card game, the expected values would be $.50, $2/3, $17/20, and $1.

Let r = # red cards remaining in deck, and b = # of black cards remaining in deck. R = B = 26 = number of reds and blacks in the deck. +1 per red, -1 per black, so current accumulated score at any point in the game is (R-r) - (B-b). Much like an American option, I'll denote the value of the game as V(r,b) and expected payout on the margin as E(r,b), so V(r,b) = (R-r) - (B-b) + E(r,b), and therefore E(r,b) needs to follow these recursive rules:

=0 if r=0
=r if b=0
=max(0, r/(r+b) * (1 + E(r-1,b)) + b/(r+b) * (-1 + E(r,b-1))) otherwise

I'm sure you can see how this sort of logic also relates to the Pascal's Triangle we just worked with :P

So I can make a table with the relevant values, and I can see that if the marginal value E(r,b) is 0, I want to quit immediately (the only real exception is when r=1 and b=2 because then r/(r+b) * (1 + E(r-1,b)) + b/(r+b) * (-1 + E(r,b-1)) = 0, so technically you'd be indifferent if you were risk-neutral, although a risk-averse player would quit).

When you're playing optimally, you never quit after drawing a black card -- only after drawing a red, and of course you can never have a negative score because drawing the whole deck yields +26 - 26 = 0 points accumulated and final value.

[leonid]

Re: Oops I broke the laws of math.

That problem is well below the difficulty of 15-251, isn't it.

Not entirely. You will have to use hard induction a LOT in 15-251, though this only counts for the first half of the course.

[Flaming_Dingleberry]

Re: Oops I broke the laws of math.

MrRubix: Explain how to fold a piece of paper into thirds (exactly) without using any measuring tools other than the paper itself.

[MrRubix]

Re: Oops I broke the laws of math.

MrRubix: Explain how to fold a piece of paper into thirds (exactly) without using any measuring tools other than the paper itself.

Uh I always just kinda rolled the bottom up into a random spot on the paper and folded the top over until it met the end of the fold -- have always been able to divide paper into thirds that way, lol.

[qqwref]

Re: Oops I broke the laws of math.

There're actually a bunch of really neat ways to divide a square piece of paper into 1/nths (for n odd) exactly. Hint for one of them: suppose the paper is already divided into 1/(n-1)ths on one side.

[stargroup]

Re: Oops I broke the laws of math.

I have concluded that aside from a select group the ffr community is incredibly retarded and unable to maintain a 5-second attention span

[Flaming_Dingleberry]

Re: Oops I broke the laws of math.

Uh I always just kinda rolled the bottom up into a random spot on the paper and folded the top over until it met the end of the fold -- have always been able to divide paper into thirds that way, lol.

Well, make a mathematical solution out of it. Like fold it in half and then fold one corner until it meets the edge of the half, unfold and fold it upwards, which will make a crease at exactly one-third of the paper's length, fold over the one-third mark, and fold again, boom, solved. Of course that makes no sense, but do something like that to get the solution.

[Ghakimx]

Re: Oops I broke the laws of math.

I have concluded that aside from a select group the ffr community is incredibly retarded and unable to maintain a 5-second attention span

FFR = Dumb

[Dorby]

Re: Oops I broke the laws of math.

2 + 2 = 5

gg song

[MrRubix]

Re: Oops I broke the laws of math.

Unfortunately, SO many people on FFR are just... scary. It's impossible to have a logical conversation with most people on this site for more than 5 minutes without feeling the need to tear my eyes out.