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Guest Message by DevFuse

More colorful foreheads

Started by bushindo, Feb 01 2013 07:34 PM

Best Answer k-man, 08 February 2013 - 03:47 PM

I think I finally got it!

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15 replies to this topic

#11 k-man

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Posted 07 February 2013 - 07:56 PM

Sure.
Spoiler for D wins
Pocket stamps are in parenthesis. in all below combinations D wins in the first round. All have equal probability of 1/126.
A  B  C  DBB BB BR RR (RR)BB BB RR RR (BR)BB BR BB RR (RR)BB RR BB RR (BR)RR RR BR BB (BB)RR RR BB BB (BR)RR BR RR BB (BB)RR BB RR BB (BR)BR BB BB RR (RR)BR RR RR BB (BB)
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Spoiler for

A B C D
BB BB BR RR (RR) *
BB BB RR RR (BR) *
BB BR BB RR (RR)
BB RR BB RR (BR) *
RR RR BR BB (BB) *
RR RR BB BB (BR) *
RR BR RR BB (BB)
RR BB RR BB (BR) *
BR BB BB RR (RR)
BR RR RR BB (BB)

You may want to check the distribution with the the star next to them. For instance, for the second distribution

BB BB RR RR (BR)

On his turn, B will reason that his hat is either BB or BR. If A saw [BB *BR* RR RR (BB)], he would have called his hat at turn 1. But A didn't, so B knows his hat.

Let's look at the first distribution, [ BB BB BR RR (RR) ]. On his turn, C reasons that his hat is either RR or BR. If it is RR, then B would have won on turn 2 due to the reasoning for distribution 2. There on his turn C would know his color.

You're right. I messed up. Here are the corrected probabilities...

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#12 bushindo

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Posted 07 February 2013 - 08:17 PM

You're right. I messed up. Here are the corrected probabilities...
Spoiler for

A: 19/63
B: 32/63
C: 10/63
D: 2/63

Based on these odds, B is still the pest position with the expected payout of almost $1.27.

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#13 k-man

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Posted 07 February 2013 - 11:42 PM

You're right. I messed up. Here are the corrected probabilities...
Spoiler for

A: 19/63
B: 32/63
C: 10/63
D: 2/63

Based on these odds, B is still the pest position with the expected payout of almost $1.27.

Comments
Spoiler for

A B C D

BB BR BB RR (RR)
RR BR RR BB (BB)
BR BB BB RR (RR)
BR RR RR BB (BB)

So it seems that the sequences that lead to D winning (probability 2/63) are above. But there are a few more sequences that would lead to D winning. For instance, consider the following distributions

BR RR BB BR BR.

There are more missing distributions for D, so you might want to look for the winning cases for A, B, C and find them.

Doh! :duh:

I went through everything again and found a couple of mistakes, so now the odds and expected payouts are as follows...

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#14 bushindo

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Posted 08 February 2013 - 02:59 AM

You're right. I messed up. Here are the corrected probabilities...
Spoiler for

A: 19/63
B: 32/63
C: 10/63
D: 2/63

Based on these odds, B is still the pest position with the expected payout of almost $1.27.

Comments
Spoiler for

A B C D

BB BR BB RR (RR)
RR BR RR BB (BB)
BR BB BB RR (RR)
BR RR RR BB (BB)

So it seems that the sequences that lead to D winning (probability 2/63) are above. But there are a few more sequences that would lead to D winning. For instance, consider the following distributions

BR RR BB BR BR.

There are more missing distributions for D, so you might want to look for the winning cases for A, B, C and find them.

Doh!

I went through everything again and found a couple of mistakes, so now the odds and expected payouts are as follows...
Spoiler for
A: 5/21 $0.83
B: 4/9 $1.11
C: 10/63 $ 0.95
D: 10/63 $ 1.11

According to this result positions B and D are equal in the expected payout, but this makes me think that I'm still missing something.

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#15 k-man

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Posted 08 February 2013 - 03:47 PM Best Answer

I think I finally got it!

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#16 bushindo

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Location:Los Angeles, CA

Posted 08 February 2013 - 04:25 PM

I think I finally got it!

Spoiler for answer

D is the best position with approx $1.11 expected payout. The odds and payouts for each position are (rounded):

A: 19/63 $1.06

B: 8/21 $0.95

C: 10/63 $0.95

D: 10/63 $1.11