Happy New Year to all of the EIU family and friends from eiumath.
To start the new year we propose a little riddle.
Two friends, both fond of disputes, are walking down the street arguing about who has the most money in their wallet, though out of sheer stubbornness neither is willing to divulge the amount of money they are carrying. They meet a friend along the way who agrees to check each wallet to see if the amounts are different.
After checking, the friend announces that one wallet has twice as much as the other. He then hands the wallets back and disappears. The two disputants realize that the wallets are identical and they are not sure whether they have the right wallets.
Then it occurs to them that by switching wallets the amount of money that each of the two friends expect to have would increase. Is this right? How is this possible? This is sometimes called the wallet game and is attributed to Maurice Kraitchik.
Subscribe to:
Post Comments (Atom)
The trick here is the relation between the two possible expected amounts for each of the two friends. Let's call the two friends Maurice and Martin.
ReplyDeleteLet M represent the amount in Maurice's wallet. Then Maurice expects to obtain 2M dollars half the time and M/2 dollars half the time. In other words on average he expects to get M + M/4 dollars by switching wallets. Thus his expectation is slightly larger than the amount in his wallet.
One can simulate this exchange as a game having two players. Suppose Maurice always carries 100 dollars to the exchange and Martin flips a coin each day so that when the coin is heads he carries 200 dollars in his wallet and when the coin is tails he carries 50 dollars.
Of course, in this simulation Martin's expectation is always 100, while Maurice's expectation is 125. The average amount of money carried to each exchange in this case is (200+100 + 50+100)/2 or 225.