A person starting with makes eight bets and wins exactly four times. The wins and losses occur in random order. If each wager is for half the money she has at the time of the bet, how much money will she win or lose?
Source: NCTM Mathematics Teacher, October 2006
Suppose the eight bets result in the following arbitrary sequence of wins and losses
The gambler loses .
What if we loaded all the four wins consecutively in the beginning so that she wins big and loses little money?
Alas! The wins are big and so are the losses. She still loses , but why lose the same amount in the two different scenarios?
Let = money available before a bet. After a win she will have more money, , that is as much as before; after a loss she will have less money , that is as much. We track the ups and downs of money using this multiplicative rule as follows
She starts out with and ends up with
Though the wins and losses occur in random order, she always loses .
Answer: Will lose