Overview and Objective
In this lesson, students will use the tools on Polypad to flip 3 coins and explore the probability of the coins landing all heads and the probability of the coins landing all tails (or stars on the Polypad coins). After making a prediction, students will flip 25 sets of 3 coins 8 different times to generate 200 trials of flipping the 3 coins. After each set of 25 flips of the 3 coins, students will calculate the percentage of all heads and all tails and graph those results on a line chart.
To introduce this lesson, have 3 coins ready to flip. Actual coins will work great, but you can also use this Polypad. Click here to learn more about using the Probability and Data tools on Polypad. Ask students to predict how many times you'll need to flip all three coins so that all the coins end up the same (either all HHH or all TTT). Invite students to share their predictions with a classmate before having some students share with the class. Flip the coins until you flip either HHH or TTT and see if any students were correct in their predictions. As time allows, do this a few more times. Now is not the time to determine the probability of flipping HHH or TTT - this will come later in the lesson. End the warm up by asking students to record a prediction of what percent of time they thing three coins will land HHH and what percent of times they think three coins will land TTT.
Share this Polypad with students. Click here to learn how to share Polypads with students and how to view their work. Using the Polypad, students will flip 25 sets of 3 coins a total of 8 times. Each time, they'll update the total number of HHH and TTT flipped and the percentage of each. You may want to do a few examples with the class before sharing the Polypad with them. The video below demonstrates the process.
As students work and you check-in with students, remind them to save their Polypad so you can view the work. Pay careful attention to those students who have great variability in their data so you can share their results with the class later on. Perhaps some students flipped very few HHH or TTT in the beginning, but flipped more at the end. Or perhaps some students flipped a good deal of them only in the beginning. Click here to see one example of a completed Polypad. If some students finish quickly, invite them to re-create this exploration with 4 coins.
When students are done, gather as a class and have each student report out their percentage out of 200 for HHH and TTT. Record all these results either on the board or on a spreadsheet so you can easily find the mean percentage for each. Be clear to state that since each percentage is out of the same number of rolls, it is accurate to find the mean of the percentages. Multiply the number of students by 200 to find the total number of flips in 3 coins and record the following (this example is with 25 students in the class):
- In 5000 flips of 3 coins, we flipped HHH ____% of the time.
- In 5000 flips of 3 coins, we flipped TTT ____% of the time.
Ask students how they think this experimental probability compares to the expected theoretical probability of flipping HHH and flipping TTT. Ask them to rank it on a scale of 1-10 with 1 being the farthest away from the theoretical probability and 10 being exactly the same as the theoretical probability. You could record their results in a table on Polypad and graph the results. This could be a good opportunity to revisit Box and Whisker plots if students have worked with these before.
Now transition to finding the theoretical probability of flipping a HHH and of flipping a TTT. Ask student to flip the 25 sets of 3 coins again and record ask student to share the different combinations they see. List them on the board as students share. Your final list should contain all 8 options:
- HHH, HHT, HTH, HTT, TTT, TTH, THT, THH
Discuss as a class that the key question now is whether or not these outcomes are equally likely. Creating a probability tree on Polypad may help students see that these are indeed equally likely outcomes. Here is one way to create the tree. Engage students in discussion and questioning as you create the tree.
Now that you've shown each outcome is equally likely, you can conclude that the theoretical probability of flipping HHH is 12.5% and the theoretical probability of flipping TTT is also 12.5%. Compare this to the whole class results and discuss how close you may or may not be to the expected probability. Finally, ask some students to share times throughout their exploration in which they were very far away from the theoretical probability. Discuss why the randomness of flipping coins can cause this to happen when the total number of trials, or sample size, is small. However, when the sample size becomes large, like 5000 in a class of 25 students, the experimental probability should become closer and closer to the theoretical probability.
Pose the following question to students:
You are playing a game with a friend in which you'll flip 3 fair coins. Player One will win if exactly two of the coins land heads. Otherwise, Player Two wins. Which player do you want to be if you want to have the greatest chance of winning?
In answering this question, students will need to examine all the equally likely outcomes in flipping 3 coins and determine that the probability of the three coins landing on exactly two heads is less than the probability of the alternative option.