70 replies to this topic

[Thalia]

100%

Spoiler for 2 solutions that work but probably aren't what you are looking for.

1. Take the cap off and look at it. I didn't see anything in the rules against it.
2. If the cap has the thing that sticks out in front, you might be able to see the color if you look up.

Edited by Thalia, 08 December 2010 - 01:15 AM.

[phillip1882]

firstly there are 15 players, not 16, and secondly..

Spoiler for

0 1 (blue)
1 - (abstain)
2 0 (red)
3 1 (blue)
4 -
5 0 (red)
etc....
okay

0 blue:
everyone guesses blue: lose
1 blue:
all red abstain, the blue guesses blue, win.
2 blue:
both blues abstain, all reds guess red. win
3 blue:
all blues guess red, all reds guess blue, lose.
4 blue:
all reds abstain, all blues guess blue, win
5 blue:
all blues abstain, all reds guess red win.
so in this case i agree, 2/3 of the time the players win.
my previous post was in response to his first strategy, even 0- 6 blue, even 8-14 red, odd abstain.

Edited by phillip1882, 08 December 2010 - 01:40 AM.

[puzzled_again]

firstly there are 15 players, not 16, and secondly..

Spoiler for

0 1 (blue)
1 - (abstain)
2 0 (red)
3 1 (blue)
4 -
5 0 (red)
etc....
okay

0 blue:
everyone guesses blue: lose
1 blue:
all red abstain, the blue guesses blue, win.
2 blue:
both blues abstain, all reds guess red. win
3 blue:
all blues guess red, all reds guess blue, lose.
4 blue:
all reds abstain, all blues guess blue, win
5 blue:
all blues abstain, all reds guess red win.
so in this case i agree, 2/3 of the time the players win.
my previous post was in response to his first strategy, even 0- 6 blue, even 8-14 red, odd abstain.

Yes, that's what I got but

Spoiler for

in reference to there only being 15 players not 16, you will find working through the above that I was actually referring to the number of possible scenarios, not players, ie. 0 blue hats through to 15 blue hats = 16 possible outcomes

[deboriole]

Spoiler for Can each person...

Write down the color of the hat of the person next to them and then pass each note one to the left?

[vinays84]

Spoiler for OK, I'll get the ball rolling at 75%

Key to a lot of these types of problems: Make it simple, then expand.
So consider all but 3 people abstain from answering.
These 3 can follow the simple strategy:
1. If the other two colors differ then abstain.
2. If the other two colors are the same, choose the opposite.

With 3 people you have 2^3=8 possibilities:
BBB GBB
BBG GBG
BGB GGB
BGG GGG
For 2/8 (BBB, and GGG) all three will guess incorrectly.
For the other 6, two will abstain and one will guess correctly.
There's your 75%.

Now, I started simple and picked 3. Maybe someone can build on this by take into people into account to raise the percentage?

[KlueMaster]

It seems all the respondents here are treating the coin toss as random event.
A fair coin toss would follow the binomial distribution for probability of a predefined pattern in successive tosses.
So, let's say what are the odds of seeing 7 reds and 7 blues? Coz that's the only 50-50 possibility for the chosen leader to guess wrong.
Whenever this balance is tilted in favor of one color, the leader should chose the other color, to account for balancing nature of binomial distribution.

[Kornrade]

It seems all the respondents here are treating the coin toss as random event.
A fair coin toss would follow the binomial distribution for probability of a predefined pattern in successive tosses.
So, let's say what are the odds of seeing 7 reds and 7 blues? Coz that's the only 50-50 possibility for the chosen leader to guess wrong.
Whenever this balance is tilted in favor of one color, the leader should chose the other color, to account for balancing nature of binomial distribution.

"The colour of the cap is determined completely arbitrarily, e.g. by coin toss."
Coin toss is only an example of random event that may be used to determine the colour on each cap independently of the other colours. Thus, all configurations from RRR...R to BBB...B are equally probable.

[Kornrade]

See next post...

Edited by Kornrade, 08 December 2010 - 09:27 AM.

[Kornrade]

Spoiler for OK, I'll get the ball rolling at 75%

Key to a lot of these types of problems: Make it simple, then expand.
So consider all but 3 people abstain from answering.
These 3 can follow the simple strategy:
1. If the other two colors differ then abstain.
2. If the other two colors are the same, choose the opposite.

With 3 people you have 2^3=8 possibilities:
BBB GBB
BBG GBG
BGB GGB
BGG GGG
For 2/8 (BBB, and GGG) all three will guess incorrectly.
For the other 6, two will abstain and one will guess correctly.
There's your 75%.

Now, I started simple and picked 3. Maybe someone can build on this by take into people into account to raise the percentage?

Well done so far!

Spoiler for That was the easy part,

going from the trivial 50% to the "simple" 75%. Now start the hard part: how to extrapolate?
Key question: what is new in this strategy compared to all other strategies suggested?

Btw, the winning percentage is well over 75%, but I shall say no more

[rajat_magic]

Spoiler for OK, I'll get the ball rolling at 75%

Key to a lot of these types of problems: Make it simple, then expand.
So consider all but 3 people abstain from answering.
These 3 can follow the simple strategy:
1. If the other two colors differ then abstain.
2. If the other two colors are the same, choose the opposite.

With 3 people you have 2^3=8 possibilities:
BBB GBB
BBG GBG
BGB GGB
BGG GGG
For 2/8 (BBB, and GGG) all three will guess incorrectly.
For the other 6, two will abstain and one will guess correctly.
There's your 75%.

Now, I started simple and picked 3. Maybe someone can build on this by take into people into account to raise the percentage?

We have to be careful with how we calculate probabilities on this. If each individual cap colour was decided by a coin toss as per the OP, then these probabilities would work. However, Kornrade has since clarified that "all configurations from RRR...R to BBB...B are equally probable." If we start from this position, then the chance of any 3 caps having the same colour is actually 50%. If the chance of each distribution was based on binomial probabilities (i.e. in the coin toss scenario), then your result would be correct, as the chance of 3 caps of the same colour would only be 25%. Attaching a spreadsheet just to illustrate the difference:

Attached Files

•  Binomial vs uniform distribution.xls   22K   70 downloads

Edited by rajat_magic, 08 December 2010 - 10:55 AM.