In pairs we played the following game a couple of times.
One person is heads (player 1), the other person is tails (player 2). Toss the coin a maximum of 10 times, player 1 gets a point if the coin lands on heads, and player 2 gets a point if the coin lands on tails. If one person becomes three points ahead of the other the game stops and they win. If no one is 3 points ahead after ten tosses the game is a tie.
Which of the following scores are impossible in this game and why? 5-2, 7-4, 6-2
Think of ways the game could end in a tie.
What is the chance of winning this game?
This following table will help.
Fill in the table shading out impossible final scores, highlighting winning final scores for Player 1 and Player 2, and highlighting final scores which indicate a draw.
Then write in each box how many ways there are to end up at each score.
A few boxes have been filled in – for example 3-0 is a winning score for player 1, and there is only 1 way to get to that score (1-0 then 2-0 then 3-0). There are two ways to get to a score of 1-1 (0-1 then 1-1 or 1-0 then 1-1). There is a 1 in the 0-0 box as there is just one way to start the game.
See here for the solution, contained in an excellent presentation by Alex in the Further Maths class in PAL. This puzzle is the starting point for analysing how likely you are to win a tennis match if you only have a one third chance of winning each point. The idea and diagrams in the presentation were taken from the book Game, Set and Math by Ian Stewart.