Dice and Probability

Explore probability with a single die

Step 1: Roll one die

1. Roll a single die 10 times.
2. Record each result using tally marks.
3. Count how many times each number appears.

Step 2: Calculate probabilities

1. Find the experimental probability of each number by dividing the number of times it occurred by the total number of rolls.
2. Compare your results with the theoretical probability of rolling each number, which is 1 in 6.

Step 3: Collect more data

1. Repeat the experiment until you have completed 60 rolls.
2. Update your tally chart after every 10 rolls.
3. Observe how the experimental probabilities change as you collect more data.

Explore probability with two dice

Step 1: Create a sample space

1. Label the columns of a 6 × 6 grid with the numbers 1 to 6 for the first die.
2. Label the rows with the numbers 1 to 6 for the second die.
3. Fill in each cell by adding the values from the row and column.
4. The completed grid shows all 36 possible outcomes when two dice are rolled.

Step 2: Roll the dice

1. Roll both dice together.
2. Add the two numbers.
3. Record the sum.
4. Repeat until you have completed 60 rolls.

Step 3: Analyse the results

1. Count how many times each sum occurred.
2. Calculate the experimental probability for each sum.
3. Compare your results with the theoretical probabilities shown by the sample space.

Investigate the Monty Hall problem

Step 1: Set up the game

1. Place a small prize under one of three cups.
2. Mix the cups so that the prize location is hidden.
3. Ask a player to choose one cup.

Step 2: Reveal an empty cup

1. Without revealing the prize, lift one of the two unchosen cups that does not contain the prize.
2. Leave the chosen cup and one other cup unopened.

Step 3: Make a decision

1. The player can either keep their original choice or switch to the other unopened cup.
2. Record whether the player wins.
3. Repeat the game multiple times.
4. Try one set of trials where the player always switches and another where the player never switches.

How does it work?

Probability describes how likely an event is to occur.

For a single die:

• There are six possible outcomes.
• Each outcome is equally likely.
• The theoretical probability of rolling any specific number is 1 in 6.

When you roll a die many times, the experimental probability usually becomes closer to the theoretical probability. This is why collecting more data often improves the accuracy of your results.

Rolling two dice produces 36 possible combinations. However, not all sums are equally likely.

For example:

• A sum of 2 can only be made in one way: 1 + 1.
• A sum of 7 can be made in six different ways.
• A sum of 12 can only be made in one way: 6 + 6.

This means a sum of 7 is much more likely than a sum of 2 or 12.

The Monty Hall problem demonstrates conditional probability.

When you first choose a cup, there is a 1 in 3 chance you picked the prize and a 2 in 3 chance you did not.

This means that switching cups gives a higher chance of winning than staying with your original choice.

Science fair project: Take it further

Once you have explored probability with dice and the Monty Hall problem, investigate how different games and sample spaces affect the results.

• Repeat the experiment using 8-sided or 12-sided dice. How do the probabilities change?
• Roll two dice 100 or 200 times. Do your experimental probabilities get closer to the theoretical probabilities?
• Compare the distributions of sums from two dice and three dice. Which sums occur most often?
• Investigate probability using a deck of playing cards. What is the probability of drawing a heart, a face card or an ace?
• Create a spreadsheet to record large numbers of trials and graph your results.
• Create a table of experimental and theoretical probabilities. Which outcomes showed the biggest differences? How did those differences change as the number of trials increased?