Real Mathematics: What are the chances?! #1

In 1987, Thomas M. Cover from Stanford University published an article which contained a bizarre result. Cover suggested the following question:

Player A and Player B will get into a competition.

Player A will be writing down distinct numbers on two pieces of papers, while Player B will be choosing one of the papers and will be reading the number written on it. Then Player B will have to decide if the bigger number is on his/her paper, or on the other one.

Naturally, Player B’s chance of finding the bigger number is 50%. In fact knowing one of the numbers has no use for his chance… or not?

Magic of Mathematics

Is there a strategy for Player B to use so that he/she will have more than 50% chance to win the game?

Although it sounds unlikely, there really is a strategy which Player B could use to win the game with a probability that is greater than 50%.

Strategy: Player B thinks of a number T on his/her mind. Every time Player B selects a paper, he/she uses T for comparison.

• If T is greater than the selected number, Player B chooses the other paper.
• If T is smaller than the selected number, Player B chooses the selected paper.

Theory

Let’s assume x and y are the numbers that are written on the papers. (x is greater than y) There are exactly three different scenarios for the game:

1. T is smaller than x and y. In this case Player B has exactly 50% chance of winning the game.
   
2. T is greater than both x and y. In this case Player B again has 50% chance of winning the game.
   
3. T is greater than y, less than x. In this case if Player B picks y, since T is greater than y, he/she will be choosing the other paper. And if Player B picks x, since T is less than x, he/she will be choosing the paper he/she picked. In both situations Player B wins the game which gives Player B a 100% chance of winning the game.
   

An Example

I will try to explain these three scenarios with an example.

Assume T is 80 and x and y are;

1. 120 and 287,
2. 1 and 2,
3. 64 and 15000.

Solutions:

1. Player B picks either 120 or 287. Since both numbers are greater than T (which is 80), Player B sticks with the paper he/she picks. In this case probability of picking 287 is exactly 50%.
2. Player B picks either 1 or 2. Since both numbers are less than T, Player B claims the other paper has the greater number on it. In this case probability of picking 1 is again exactly 50%.
3. Player B picks either 64 or 15000. T is between these numbers. If Player B picks 64, since it is less than T, Player B would pick the other paper. If Player B picks 15000, since it is greater than T, Player B would claim that the paper he/she picked is the greater number. In both cases Player B wins, which gives him/her a probability of 100% for winning the game.