
Re: (difficult)Theoretical gambling puzzle (solution?)
Posted:
Jul 26, 2006 7:42 AM


Hi,
Here are the initial values given by my method:
2+2 => 2/3 3+3 => 17/20 4+4 => 1 5+5 => 47/42 6+6 => 284/231 7+7 => 4583/3432 8+8 => 18457/12870 9+9 => 74131/48620 10+10 => 26995/16796
 Eric
At 08:13 26/07/2006, JoÃ£o Pedro Afonso wrote: >Hi Eric, > > My method is very similar to yours (but not > exactly equal) and I don't understand why it > doesn't give the same values, yet. But I > didn't found any problems in the reasoning, so, > maybe the small diferences are significative > and you have a better stop criteria. Can you > send the expected values for 2+2, 3+3, and 4+4. > To 2+2, it is easy to see it must be 2/3. > > > Cheers, >Joao Pedro Afonso > > >Eric Wrote: > > Hi again and again :) > > > > Well, maybe I should stick to geometry... > > My current value is 2.6244755... (seems to converge > > towards yours :) > > > > I assumed that the expected gain E(P,N) depends only > > on the quantities > > of red (P for positive) /black (N for negative) cards > > remaining in the deck. > > > > E(0,N) is N (no more positive, stop now) > > E(P,0) is 0 (no more negative, draw all remaining > > cards to return to 0) > > > > In the general case, we compare the current gain S = > > (NP) if we stop now > > to the expected gain if we continue C = > > E(P1,N)*P/(N+P) + E(P,N1)*N/(N+P). > > Here I have a little doubt about the probability of > > picking a P or a N, then > > E(P,N) = max(S,C) > > > >  Eric > > > > At 12:39 25/07/2006, Joao Pedro Afonso wrote: > > > > > Hi Eric, :) > > > > > > I think you escaped from a good one. Now I have > > to read the > > > mapple script, but look what I was preparing to > > send in reply to > > > your first message, in the moment it arrived to the > > MathOrg (where > > > it is probably waiting now for the moderators > > approval): > > > > > >:) > > > > > >Eric Bainville wrote: > > >>Hi, > > >>Very fun problem! > > >>My strategy gives an expected gain of > > >>805867616040669 / 281474976710656 = 2.863016903... > > > > > > Grrrr! Your expected gain is bigger than mine!!! > > >( > > > > > >>Small numbers of cards can be checked by hand (P is > > a +1 card, N is > > >>a 1 card). > > >... > > >>2+2 cards: > > >>P: continue => > > >> PP: stop 2 (1/4) > > >> PN: continue => > > >> PNP: stop 1 (1/8) > > >> PNN: continue => PNNP 0 (1/8) > > >>N: continue => > > >> NP: continue => > > >> NPP: stop 1 (1/8) > > >> NPN: continue => NPNP 0 (1/8) > > >> NN: continue => NNPP 0 (1/4) > > >>=> 3/4 > > > > > > Hufff! Look careful the way you are doing your > > probabilities. :) > > > > > > > > > Cheers, > > >Joao Pedro Afonso > > >

