## Welcome to BrainDen.com - Brain Teasers Forum

 Welcome to BrainDen.com - Brain Teasers Forum. Like most online communities you must register to post in our community, but don't worry this is a simple free process. To be a part of BrainDen Forums you may create a new account or sign in if you already have an account. As a member you could start new topics, reply to others, subscribe to topics/forums to get automatic updates, get your own profile and make new friends. Of course, you can also enjoy our collection of amazing optical illusions and cool math games. If you like our site, you may support us by simply clicking Google "+1" or Facebook "Like" buttons at the top. If you have a website, we would appreciate a little link to BrainDen. Thanks and enjoy the Den :-)
Guest Message by DevFuse

# brob26

Member Since 13 Aug 2013
Offline Last Active Aug 15 2013 10:26 PM

### In Topic: Prisoners and hats redux

15 August 2013 - 07:31 PM

Spoiler for

### In Topic: i am thinking of a number...

14 August 2013 - 12:42 PM

Spoiler for solution

### In Topic: Colored cards at Morty's

13 August 2013 - 10:55 PM

Starting with r reds and b blacks, the expected return is

V(r,b) = 2r+b/(r+bCr)

Note that this satisfies

V(r,b) = (r/r+b)*V(r-1,b) + (b/r+b)*V(r,b-1)

Therefore V is a martingale! This means that ANY rational strategy will yield the same expected return. 'Rational' here just means never betting on the rarer colour, and always betting everything when only one colour remains.

Moreover, there exists a risk-free strategy of always betting a proportion (r-b)/(r+b) of current wealth (or likewise (b-r)/(r+b) if there are more blacks remaining), such that one is GUARANTEED a return of \$9.08.