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Science Forum Index » Mathematics Forum » Probability with Psacal and Fermat..
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| mina_world |
 Posted: Sat Apr 26, 2008 5:11 am |
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Guest
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Hello teacher~
The problem of points :
Two playwers(A and B) of equal skill play a game.
(thik of tossing a fair coin).
The first one to win a fixed number of games (say 5) wins the whole stake.
The game is interrupted when player A wins 4 and player B wins 3.
(Namely, player A needs 1 to win and player B needs 2 to win.)
How should the stake be divided ?
----------------------------------------------------
Since player A wins 4 and player B wins 3,
the number of game is 7.
All cases of 8, 9 games
(A wins, A wins) ==> A wins
(A wins, B wins) ==> A wins
(B wins, A wins) ==> A wins
(B wins, B wins) ==> B wins
so, answer is 3 : 1.
is this really fair ?
- Lawyer Fermat and Religious Pascal - |
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| quasi |
| Posted: Sat Apr 26, 2008 5:30 am |
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Guest
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On Sat, 26 Apr 2008 19:11:24 +0900, "mina_world"
<mina_world@hanmail.net> wrote:
Quote: Hello teacher~
The problem of points :
Two playwers(A and B) of equal skill play a game.
(thik of tossing a fair coin).
The first one to win a fixed number of games (say 5) wins the whole stake.
The game is interrupted when player A wins 4 and player B wins 3.
(Namely, player A needs 1 to win and player B needs 2 to win.)
How should the stake be divided ?
----------------------------------------------------
Since player A wins 4 and player B wins 3,
the number of game is 7.
All cases of 8, 9 games
(A wins, A wins) ==> A wins
(A wins, B wins) ==> A wins
(B wins, A wins) ==> A wins
(B wins, B wins) ==> B wins
so, answer is 3 : 1.
is this really fair ?
Yes.
quasi |
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| quasi |
| Posted: Sat Apr 26, 2008 5:40 am |
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Guest
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On Sat, 26 Apr 2008 06:30:30 -0400, quasi <quasi@null.set> wrote:
Quote: On Sat, 26 Apr 2008 19:11:24 +0900, "mina_world"
mina_world@hanmail.net> wrote:
Hello teacher~
The problem of points :
Two playwers(A and B) of equal skill play a game.
(thik of tossing a fair coin).
The first one to win a fixed number of games (say 5) wins the whole stake.
The game is interrupted when player A wins 4 and player B wins 3.
(Namely, player A needs 1 to win and player B needs 2 to win.)
How should the stake be divided ?
----------------------------------------------------
Since player A wins 4 and player B wins 3,
the number of game is 7.
All cases of 8, 9 games
(A wins, A wins) ==> A wins
(A wins, B wins) ==> A wins
(B wins, A wins) ==> A wins
(B wins, B wins) ==> B wins
so, answer is 3 : 1.
is this really fair ?
Yes.
Assuming the stake is x dollars,
the expected value for player A is
(3/4)*x + (1/4)*0 = (3/4)*x
and the expected value for player B is
(1/4)*x + (3/4)*0 = (1/4)*x,
so the fair division of the stake is in the ratio 3:1.
quasi |
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| mina_world |
| Posted: Sat Apr 26, 2008 6:12 am |
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Guest
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"quasi" <quasi@null.set> wrote in message
news:6d16141b2v7l8t04kv930bif89abuf4q4j@4ax.com...
Quote: On Sat, 26 Apr 2008 06:30:30 -0400, quasi <quasi@null.set> wrote:
On Sat, 26 Apr 2008 19:11:24 +0900, "mina_world"
mina_world@hanmail.net> wrote:
Hello teacher~
The problem of points :
Two playwers(A and B) of equal skill play a game.
(thik of tossing a fair coin).
The first one to win a fixed number of games (say 5) wins the whole
stake.
The game is interrupted when player A wins 4 and player B wins 3.
(Namely, player A needs 1 to win and player B needs 2 to win.)
How should the stake be divided ?
----------------------------------------------------
Since player A wins 4 and player B wins 3,
the number of game is 7.
All cases of 8, 9 games
(A wins, A wins) ==> A wins
(A wins, B wins) ==> A wins
(B wins, A wins) ==> A wins
(B wins, B wins) ==> B wins
so, answer is 3 : 1.
is this really fair ?
Yes.
Assuming the stake is x dollars,
the expected value for player A is
(3/4)*x + (1/4)*0 = (3/4)*x
and the expected value for player B is
(1/4)*x + (3/4)*0 = (1/4)*x,
so the fair division of the stake is in the ratio 3:1.
What's the criterion of "FAIR" ? |
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| quasi |
| Posted: Sat Apr 26, 2008 6:16 am |
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Guest
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On Sat, 26 Apr 2008 20:12:48 +0900, "mina_world"
<mina_world@hanmail.net> wrote:
Quote:
"quasi" <quasi@null.set> wrote in message
news:6d16141b2v7l8t04kv930bif89abuf4q4j@4ax.com...
On Sat, 26 Apr 2008 06:30:30 -0400, quasi <quasi@null.set> wrote:
On Sat, 26 Apr 2008 19:11:24 +0900, "mina_world"
mina_world@hanmail.net> wrote:
Hello teacher~
The problem of points :
Two playwers(A and B) of equal skill play a game.
(thik of tossing a fair coin).
The first one to win a fixed number of games (say 5) wins the whole
stake.
The game is interrupted when player A wins 4 and player B wins 3.
(Namely, player A needs 1 to win and player B needs 2 to win.)
How should the stake be divided ?
----------------------------------------------------
Since player A wins 4 and player B wins 3,
the number of game is 7.
All cases of 8, 9 games
(A wins, A wins) ==> A wins
(A wins, B wins) ==> A wins
(B wins, A wins) ==> A wins
(B wins, B wins) ==> B wins
so, answer is 3 : 1.
is this really fair ?
Yes.
Assuming the stake is x dollars,
the expected value for player A is
(3/4)*x + (1/4)*0 = (3/4)*x
and the expected value for player B is
(1/4)*x + (3/4)*0 = (1/4)*x,
so the fair division of the stake is in the ratio 3:1.
What's the criterion of "FAIR" ?
How much each player would win, on average, if it were played out a
large number of times.
quasi |
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| quasi |
| Posted: Tue Apr 29, 2008 3:53 am |
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Guest
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On Mon, 28 Apr 2008 17:30:40 -0700 (PDT), bill <b92057@yahoo.com>
wrote:
Quote:
On Apr 27, 11:12 am, quasi <qu...@null.set> wrote:
On Sun, 27 Apr 2008 18:39:53 -0700 (PDT), bill <b92...@yahoo.com
wrote:
On Apr 27, 5:59 pm, quasi <qu...@null.set> wrote:
On Sun, 27 Apr 2008 15:12:10 -0700 (PDT), bill <b92...@yahoo.com
wrote:
On Apr 26, 3:11 am, "mina_world" <mina_wo...@hanmail.net> wrote:
Hello teacher~
The problem of points :
Two playwers(A and B) of equal skill play a game.
(thik of tossing a fair coin).
The first one to win a fixed number of games (say 5) wins the whole stake.
The game is interrupted when player A wins 4 and player B wins 3.
(Namely, player A needs 1 to win and player B needs 2 to win.)
How should the stake be divided ?
----------------------------------------------------
Since player A wins 4 and player B wins 3,
the number of game is 7.
All cases of 8, 9 games
(A wins, A wins) ==> A wins
(A wins, B wins) ==> A wins
(B wins, A wins) ==> A wins
(B wins, B wins) ==> B wins
so, answer is 3 : 1.
is this really fair ?
Not really. A should get 4/7 of the stake and B should get 3/7.
By what logic?
quasi
The division should be based on the
number of games won!
So let's see your calculation.
I am assuming that 7 games were played.
I calculate that each game is worth 1/7 th of the stake
If B wins 4 games, then his share is 4/7 and B's share is 3/7
My answer is based on what "I" think is "fair".
Unless you can justify your answer logically in a way that would be
consistent for other such situations, what you "think" is fair (4/7
for A, 3/7 for B) is just a wild guess.
Moreover, it definitely _doesn't_ agree with the usual valuation
concept based on expected value.
Quote: I do not think that "all or nothing" is "fair".
Neither do I.
A is not guaranteed to win, so awarding the full stake to A deprives B
of the chance B would have had if the game had continued.
Quote: Myna hinted that the stake should be divided according to
the probability of eventually winning 5 games.
Sure, that makes sense, since if they were to play it out, that
division is how much each would win, over the long run.
Quote: But I think that the division should be based on the actual results
What "actual results" are you talking about?
Quote: rather than on the probable results!
Probable results is all we have in this situation.
Quote:
I have no idea as to what the "correct" division should be!
Why didn't you say that in the first place!
Quote: If the question of "fairness" is moot;
But it's not moot -- fairness is what this problem is all about.
Quote: I would not object to giving 100% of the stake to A.
I suspect A wouldn't object either.
On the other hand, B _should_ object.
Quote: Now that I have gone through this tirade; what do you
think that Mina was really trying to ask?
Her question seemed clear to me. She wanted to know whether the ratio
3:1 represents a fair division.
Based on the concept of expected value, the answer is "yes".
Thus, the division would be
3/4 of the stake for A
1/4 of the stake for B
As an alternate way to understand it, suppose A has to leave, but is
allowed to sell the position. The buyer C gets to take over the game
from where A left off. C How much should C pay? On average, C wins
3/4 of the time, winning the full stake, and loses 1/4 of the time,
getting nothing. Suppose C pays x times the stake for the privilege of
taking over A's position. On average, over the long run, C nets (3/4 -
x) times the stake. If x is less than 3/4, then over the long run, C
achieves a net average gain, while if x is more than 3/4, then over
the long run, C sustains a net average loss. Hence x = 3/4 is exactly
fair.
quasi |
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