Probability Activities for the Classroom With Dice and Coins

The six lessons at a glance

Lesson 1: Is a coin fair?

Ask the class: if you flip a coin 20 times, how many heads will you get? Most will say 10. Pairs flip a real coin 20 times and tally. Put every pair's result on the board. Only a few pairs get exactly 10; the chance of exactly 10 heads in 20 flips is only about 18%.

Then open the Coin Flip, choose Custom and enter 1,000. The heads count lands much closer to half, in proportion. Talk about why a small sample wanders and a big one settles. Add up every pair's tallies for a class total, which also gets closer to half.

Lesson 2: Real or fake streaks

Each student writes down a made-up sequence of 30 heads and tails, trying to make it look random. Collect them. Then each student flips a real coin 30 times and writes that down too. Mix up the papers and ask the class to sort them into real and made up.

Look at the longest run on each paper. In 30 real flips, a run of at least four of the same side happens about 90% of the time, and a run of five or more about 62% of the time. Do the made-up sequences have runs that long? Check it on screen: flip 100 coins and find the longest run in the result. The Coin Flip page has a table of how likely streaks are, for older students who want the numbers.

Lesson 3: The lucky die

Students roll one real die 30 times and tally each face. Ask: does anyone have a lucky number? The counts will rarely be even, and some faces will come up far more often than others, even with a fair die.

Now open the Dice Roller, choose d6 and set Number of dice to 20. Five presses of Roll give 100 rolls. Tally the faces shown under each total. Each face should come up close to one sixth of the time, about 17 times in 100, but rarely exactly. The lesson is the same as with coins: a fair die can look unfair in a small sample.

Lesson 4: Two dice totals

Students draw a 6 by 6 grid, label the rows and columns 1 to 6 and write the total in each square. Then they count how many squares give each total from 2 to 12. They should find that 7 appears in more squares than any other total, and 2 and 12 in only one each.

Check with the dice roller set to two d6. The history shows the last 20 rolls, so record totals after every 20 presses. After 100 rolls, compare the class tally with the grid. The Dice Roller page lists the exact chance of every total, for checking the counting.

Lesson 5: Is this game fair?

Describe a game: a number from 1 to 10 is drawn. Player A wins on 1 to 4, Player B wins on 5 to 10. Ask students whether it is fair before doing any math. Then they work it out: A wins on 4 numbers out of 10, a 40% chance; B wins on 6, a 60% chance.

Test it with the Random Number Generator: Min 1, Max 10, Quantity 100 and Allow duplicates on. Count A wins and B wins. Then ask students to change the rules so the game is fair, and test their new rules the same way.

Lesson 6: Keep the higher die

Roll two dice and keep the higher number. What is the chance it is a 6? Students use their grid from lesson 4 again, this time writing the higher number in each square. A 6 is the higher number in 11 of the 36 squares, about 31%, compared with 1 in 6, about 17%, for a single die. A 1 shows up in only one square.

Check on screen with two d6 and the history. This is the same idea as advantage in role-playing games, which the Dice Roller page explains for twenty-sided dice.

Running the lessons

• Materials. One coin and one die per pair, paper for tallies, and one device on the projector. Students do not need their own devices.
• Predict first. Always ask for a prediction before any flipping or rolling. Students remember the surprise when they were wrong.
• Pool the data. A class of 30 doing 20 flips each has 600 flips. Adding them up is the fastest way to show results settling.
• Words to use. Experimental probability is what the class measured. Theoretical probability is what the math says. The lessons are about the gap between them.

How to record an experiment so it can be checked

Before a trial, write down how many outcomes there are, how many attempts you will make and which event you are counting. For a coin, tally heads and tails separately. For two dice, decide in advance whether you are counting totals or single faces.

At the end, compare the share you observed with the expected fraction. A gap on its own does not prove that a coin or die is unfair. Repeat the experiment, or pool several groups' results, before drawing a conclusion.

Going further with older students

Grade 7 and 8 students can predict the counts before they collect them. For 100 rolls of a die, they expect about 17 of each face. After the experiment, they work out how far each count landed from the prediction and decide together how far off is too far to still call the die fair. There is no single right line, which leads to a good discussion about evidence.

Another extension: ask students to predict the chance of at least one 6 when rolling two dice. Adding one sixth and one sixth gives one third, which is a tempting answer. The grid gives 11 out of 36, and the gap is a good way into why chances do not simply add up.

Questions to end with

• Why did the screen results look fairer than ours?
• If a coin landed heads five times in a row, what is the chance of heads next?
• How could you design a game that looks fair but is not?
• Where else do people use probability: weather, sports, games?