Modeling Prompt

Arrange students in pairs and give each pair two coins. Tell students that they will play a game with these coins. Here are the rules:

• One person will be the Flipper—in all the rounds of the game, that person will flip the coins.
• If both coins land on the same side, the Flipper wins. But if one lands on heads and the other lands on tails, the other person wins.

Ask students if they think this is a fair game. That is, are both players equally likely to win? After some quiet think time, invite students to share their reasoning about whether the game is fair. (There are four possible outcomes that are equally likely, and in two of them, the Flipper wins, and in the other two, the other person wins, so this seems fair.) 

After a few have shared, invite students to test whether the game is fair by playing a few rounds and recording their results. Individual pairs do not need to play many rounds, because the results from the whole class will be used to see whether the game is fair.

Make a table to record the outcomes for all to see—for example, two columns with the headings “Flipper wins” and “Flipper loses.” When pairs have their results, invite them to record the results in the table—for example, by adding tally marks to each side. When all groups have recorded their results, ask students, “Do the results support your opinion about whether the game is fair?” (Out of the 50 rounds we played, the Flipper won 28 times and lost 22 times, which is close to a 50% chance of winning, so this supports my opinion that the game is fair.)

An important point to bring out in the discussion is that playing the game can help us see which conjectures about the probabilities are plausible, but we still need to make a mathematical argument to prove whether the game is fair, because the results we get when we actually play the game might not match the mathematical odds perfectly.

Tell students that in this task, they will decide whether certain games like the one they have just played are fair, and they will design their own fair games. If they have trouble deciding whether a game is fair, they can play it for a while and record their results in order to get a rough idea of whether it’s fair.