Three coins are lying on a table: a quarter, a half dollar, and a silver dollar. You claim one coin, I’ll claim the other two, and then we’ll toss all three.
A coin that lands tails counts zero, and a coin that lands heads wins its value
(in cents, 25, 50, or 100) for its owner.
Whichever of us has the larger score wins all three coins. If all three coins land tails then we call it a draw and toss again.
Which coin should you claim to make the game fair — that is, so that each of us has an expected win of zero?
Source: Martin Gardner, “Charles Addams’ Skier and Other Problems,” in Wheels, Life and Other Mathematical Amusements, 1983.
Well, in that case ...
Ignoring the the one time in 8 when all three coins land face down:
Whoever owns the $1 coin wins it all 4 times out of 7 (whenever it comes up heads)
Whoever owns the 50 cent coin wins it all 2 times out of 7.
Whoever owns the 25 cent coins wins it all 1 time out of 7.
If the single coin owner owns just the quarter, then he wins $1.50 1 time out of 7 and loses $.25 6 times out of 7. So it is a fair game.
If the single coin owner owns just the half dollar, then he wins $1.25 2 times out of 7 and loses $.50 5 times out of 7. So it is a fair game.
If the single coin owner owns just the dollar, then he wins $.75 4 times out of 7 and loses $1.00 3 times out of 7. So it is a fair game.
So this is a fair game no matter how the coins are divided.
For that matter, if one player owns all the coins, it is a fair game also, but no a very interesting one.