[Meowrocket]

Flippin' Jeff.

Flippin' Bones.

Flippin' double post.

Flippin' quotes.

Flappin' incorrect quotes.

Floutin' flautists.

Flippin' Coins.

[Rick]

Flippin' Jeff.

Flippin' Bones.

Flippin' double post.

Flippin' quotes.

Flappin' incorrect quotes.

Floutin' flautists.

Flippin' Coins.

Dang flabbit

[dlgn]

Flippin' Jeff.

Flippin' Bones.

Flippin' double post.

Flippin' quotes.

Flappin' incorrect quotes.

Floutin' flautists.

Flippin' Coins.

Suppose that you have $1000 and I am going to flip a fair coin. You choose a bet of x from $0 to $1000. If you win the flip, you will gain a total of 3x (so $1 would bring you up to $1003). If you lose the flip, you will lose x (so $1 would bring you down to $999). I will repeat the coin flip after, and again and again for some large but finite amount of times (you can decide what that means). You may change your bet each time, but it must always be from $0 to your total. You will no longer be able to participate if you go down to $0. How much should you bet for the first flip?

The answer is $333.33, in case you were wondering. You should continue to bet 1/3 of your total wealth each time to maximize your expected value. You can find this by optimizing the function

f: [0,n] --> R defined such that f(x) = .5(ln(1000+3x)+ln(1000-x))

which determines your probabilistic expected outcome on a logarithmic scale. You'll get 1/3 every time.

More generally, we can say that given this situation with a probability P∈[0,1] of winning Wx and a total wealth of n, the amount most efficient to bet is one of the following:

0 (as in the case of P=0), with an expected logarithmic value of ln(n), and thus an expected value of n, or

(n/w)(PW-1+P), with a logarithmic expected value of (hold onto your hats, folks) P*ln(n+n(PW-1+P))+(1-P)*ln(n-(n/w)(PW-1+P)).

You've got to plug in the numbers and calculate them manually to find out. Or maybe not, but I'm far too lazy to waste my time trying to figure out if the second is generally greater than the first outside of P=0. That expression is a nightmare. Anyway, that's your math fact of the day.

~dlgn

EDIT: When my old orchestra was on tour in Europe, we were practicing in a hot room in Vienna and one of our flautists passed out in the middle of rehearsal. So if by "flipping" you mean "passing out in the middle of rehearsal and falling off your chair onto the floor, subsequently causing everyone around you to freak out" then yes, flipping flautists indeed.

[dlgn]

again and again for a potentially infinite or at least unlimited and indefinite amount of times

FTFM. If it's finite, then technically the best strategy is to just bet all your money every time. Obviously this doesn't really work, but it's pretty basic probability theory, and doesn't take into account differences in relative value—i.e. going from 1 million to 0 is waaaay worse than going from 2 million to 1 million. But ignoring that, the potential payoff is always worth the risk, unless the game keeps going infinitely, in which case there's a theorem (with a name I can't remember) that says that you will eventually lose at least once, meaning betting 100% each time is sure to get you down to $0 eventually. Then, the optimal bet to make is as I said above.

[MindlessInsanity]

mega snip

.

You sound like a bigger krelboyne than me.

[eah]

again and again for a potentially infinite or at least unlimited and indefinite amount of times

FTFM. If it's finite, then technically the best strategy is to just bet all your money every time.

I'm afraid this will be your future:

Spoiler:

I do get it. You have a huge chance of leaving with nothing and a super small chance of leaving with a ton of cash. The average return is more than what you started with, but the typical return is 0. Still, if everyone on the planet had plenty of money (like a lot more than $1000) and were allowed to play, most would choose not to bet it all even though it is rational to do so.

Also, I'm most definitely not going to bet it all if I don't know how many times we flip before playing since the house could potentially "cheat" and flip only until I lose. Therefore, I would go home empty-handed every time.

[dlgn]

Like I said, it isn't the best model for finite flips because that version doesn't use logarithms, so it doesn't take differences in relative value into account.

[Eetrab]

Like I said, it isn't the best model for finite flips because that version doesn't use logarithms, so it doesn't take differences in relative value into account.

Out of curiosity what type of math are you studying/studied last?

[dlgn]

I'm currently taking an introductory calculus class (for credit and completeness), but I know more math from my own personal reading. I don't know for certain which precise field I want to go in but I think that algebraic geometry, number theory, and set theory all sound very interesting. Then again, so do many others, and I don't think I've taken enough classes to even know all of them yet, let alone decide which I want to specialize in.

[Eetrab]

I'm currently taking an introductory calculus class (for credit and completeness), but I know more math from my own personal reading. I don't know for certain which precise field I want to go in but I think that algebraic geometry, number theory, and set theory all sound very interesting. Then again, so do many others, and I don't think I've taken enough classes to even know all of them yet, let alone decide which I want to specialize in.

You should take statistics. I think you'd like it.