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Topic: Beating the Odds?
Replies: 35   Last Post: Feb 6, 2013 3:44 PM

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Posts: 12,067
Registered: 7/15/05
Re: Beating the Odds?
Posted: Feb 3, 2013 12:36 PM
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Here's a finite version of the problem, cast as a game.

Perhaps the game is well known, I'm not sure, however the
resolution of the game is nice, and not immediately obvious,
so I'll pose it here as a challenge.

2-player play a game, win or lose, for 1 dollar, based on a
fixed positive integer n > 1, known in advance to both players.

Player 1 chooses two distinct integers from 1 to n inclusive,
writes them on separate index cards, and places them face down
on the table.

Player 2 then selects one of the cards and turns it face up,
exposing the hidden value. Player 2 can then either "stay",
yielding the value on the chosen card, or "switch", yielding the
value on the other card instead. Player 2 wins (and player 1
loses) if player 2's final value is the higher of the 2 values,
otherwise player 2 loses (and player 1 wins).

In terms of n, find the value of the game for player 2, and
specify optimal strategies for both players.


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