Unfinished Game

An Advanced Mathematics Lesson Starter Of The Day

A coin is tossed repeatedly. If it comes up heads Pascal gets a point but if it comes up tails Fermat wins a point. The first person to win three points is the winner and receives the prize of £12.

Unfortunately the game had to end abruptly after three tosses of the coin. Pascal had two points and Fermat had one point. They decided to share the £12 in a ratio that matched the probability of them winning the game if it had continued.

How should they divide the £12?




You may be surprised at the correct answer as it is not £8 and £4!


Topics: Starter

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    The £12 should be divided in the ratio of the probabilities of the two players winning which can best be shown in a simple tree diagram.

    Alternatively consider the following equally likely scenarios for the outcome of the next two tosses:

    Heads heads

    Heads tails

    Tails heads

    Tails tails.

    Three of the four possibilities would make Pascal the winner so the prize should be shared in the ratio 3:1 meaning that Pascal should receive £9 and Fermat should receive £3.

    More Mathematics Lesson Starters

    Eric Levy, United States

    Thursday, November 17, 2016

    "Thank you for the podcasts! I really enjoy the puzzles. This relates to the 5/29/15 podcast re: coin flipping game that was stopped before completion. The flips when stopped were two Heads and one Tail. You indicate that the options on next 2 tosses are HH, TH, HT, and TT. Since the game stops when one person reaches 3 points, wouldn't HH and HT be the same, as the second flip isn't needed? This seems to match your ultimate answer of 75% and 25%, though ... with the person with Tails only winning with TT, which is 25% chance. I get the same answer but intermediate steps differ. Am I looking at this incorrectly? Thanks!"


    Thursday, November 17, 2016

    "Dear Eric, Thanks so much for your observations and you are completely correct. The only reason I chose to list the next two outcomes was to produce ‘equally likely’ outcomes making the arithmetic very slightly easier. I’m glad you enjoy the puzzles."

    How did you use this starter? Can you suggest how teachers could present or develop this resource? Do you have any comments? It is always useful to receive feedback and helps make this free resource even more useful for Maths teachers anywhere in the world. Click here to enter your comments.

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