Win or Lose?
Problem
A gambler bets half the money in his pocket on the toss of a coin, winning an equal amount for a head and losing his money if the result is a tail. He repeats this many times, each time betting half the total money he has. After $2n$ plays he has won exactly $n$ times. Has he more money, the same amount or less money than he started with?
Student Solutions
The gambler will have less money than he started with.
Suppose the amount of money before a game is $m$, then:
$m \to 3m/2$ for a win and $m\to m/2$ after losing a game.
| Values of n | Amount after 2n games: n wins, n losses |
|---|---|
| 1 | $3m/4$ |
| 2 | $m \times 1/2 \times3/2 \times1/2 \times3/2 = (3/4)^2 m$ |
| 3 | $m \times1/2 \times3/2 \times1/2 \times3/2 \times1/2 \times3/2 = (3/4)^3 m$ |
After $n$ wins and $n$ losses he will have $(3/4)^n$ times the money he started with, irrespective of the order in which his wins and losses occur. Eventually he will run out of money as what he has left will be smaller than the smallest coin in circulation.
The diagram was suggested by Roderick and Michael of Simon Langton Boys' Grammar School Canterbury who pointed out that if the gambler went on indefinitely he would, in theory, end up with an infinitely small amount which would be represented by nothing.