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Guest Message by DevFuse


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

Posts I've Made

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

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.