Favourable outcomes

We have spent a lot of time rolling dice in my MDM 4U classes so far this year. One day we played 3 different short games all involving being in pairs and rolling a pair of dice.

Here are the details of the three games.

Game #1:Roll 2 dice and consider the sum of the dice.
Player A wins 1 point if the sum is EVEN. Player B wins 1 point if the sum is ODD.

Game #2: Roll 2 dice and consider the product of the dice.
Player A wins 1 point if the product is EVEN. Player B wins 1 point if the product is ODD.

Game #3: Roll 2 dice and consider the sum of the dice.
Player A wins 2 point if the sum is EVEN. Player B wins 1 point if the sum is greater than or equal to 7.

This lead to a discussion of which player would you choose to be in the game and why.

For Game #1, I had a justification for 3 different outcomes:
1. Choose Player B because the mode sum is 7 so there would be more odd outcomes.
2. Choose either one because Even and Odd are equally likely (but not sure why they are equally likely)
3. Choose Player A because Even outcome more frequent
Even # + Even # = Even #
Even # + Odd # = Odd #
Odd # + Odd # = Even #

It was great that students were willing to share their thought process and we had supporters and opponents of each theory. We then developed a chart to show all possible outcomes of the sums and saw that both Even and Odd outcomes are equally likely. Probability was introduced.

In Game #2, based on their results of playing the game, each student would choose to be player A. Similar to game #1, by examining a chart of the possible products, it was clear that Even outcomes are more likely.

We started to discuss the idea of fair games and mutually exclusive events.

Homework for the class was to determine how to chance Game #2 so that it was equally likely for Player A or Player B to win the game. Students came up with some alternate version of one of the following:

- Player A gets 1 point for an EVEN product and Player B gets 3 points for an ODD product (changing the point value of each player)
- Player A gets 1 point for a product of 1 and Player B gets 1 point for a product of 36. All other products result in no points for either player. (only selecting some of the products)

Further discussions in upcoming classes will focus on fair games.