Minds On

Probability with dice

Student Success

Think-Pair-Share

There are three dice. Make a prediction for what the sum of all three dice will be once they are rolled. Record your predictions. Throughout this learning activity, you can record your thoughts digitally, orally, or in print.

The most common, or most probable, roll for two dice is a sum of seven. Why do you think this is the case?

Note to teachers: See your teacher guide for collaboration tools, ideas and suggestions.

Action

What is probability?

Probability means how likely an event is to occur. We can discuss the probability of events as certain, impossible, likely, or unlikely. For each event in the following multiple choice activity, select the corresponding probability you believe best fits.

Probability as fractions

The probability, or the possibility of an event occurring can be represented as a fraction.

The numerator (top of the fraction) records the number of the desired outcome and the denominator (bottom of the fraction) records the number of possible outcomes.

Formula of probability as a fraction. Title reads, “probability fraction.”

An experiment does not always need to be conducted to determine the probability fraction.

For example, a die has six possibilities (six different numbers). That means that every time you roll a die the probability of rolling a 1 is 16 (my desired outcome/six possible outcomes). If you roll the die 30 times, there are now 30 possible outcomes. The denominator is 30.

To determine the numerator, we can create a fraction that is equivalent to 16. Just because you are rolling the die more times, the chance that you roll a 1 doesn’t change. Since 6×5 is 30, you can do 1×5 to determine the numerator. Therefore, your probability fraction is 530.

Press ‘Hint’ to reveal an explanation of the steps required to create a probability fraction.

Step 1: Determine the numerator by considering how many of your chosen objects have the desired outcome.

Step 2: Determine the denominator by adding together all of the chosen objects to determine how many total possibilities there are.

Step 3: Construct your probability fraction! Can it be reduced? Use the fraction to describe how probable this event is.

Do you think 530 describes something that is very probable? How could you use words to describe the likelihood of what happened?

If you would like, you can complete the next activity using TVO Mathify. You can also use your notebook or the following fillable and printable document.

Create a fraction to represent the following scenarios:

  1. There is a box of candies with five smarties, three gummies, and four gumdrops. What is the probability that a gummie will get picked?
     5 blue smarties, 3 pink gummies, 4 green gumdrops on a table.
  2. You have a regular deck of 52 cards. There are 13 cards in each suit (13 hearts, 13 diamonds, 13 spades, and 13 clubs. There are ace, two to 10, jack, queen, and king in each suit. You will shuffle the deck and randomly pull a card. Use fractions to describe how probable each event is.
  • What is the probability that you pull a 5 of spades or clubs?
  • What is the probability that you pull a 5 of hearts?

Press the ‘TVO Mathify' button to access this interactive whiteboard and the ‘Activity’ button for your note-taking document. You will need a TVO Mathify login to access this resource.

TVO Mathify (Opens in new window) Activity (Open PDF in a new window)

Probability lines

Probability lines can be used to explain how probable something is based on its fraction.

On a probability line, probabilities are represented anywhere from zero to one.

  • zero means that an event has no chance of happening or is impossible (0%).
  • one means that the event is for sure going to happen or is certain (100%).

All of the other probabilities are represented with fractions or decimals in between zero and one. They can be easily turned into percentages that describe how likely an event is by multiplying the decimal by 100.

Probability line with 0 at one end and 1 at the other, broken in 4 intervals of 0.25, 0.5, and 0.75.

The previous picture shows a probability line with 0 at one end and 1 at the other. It is broken into four intervals of 0.25, 0.5, and 0.75. If zero is impossible and is one certain, what numbers could represent unlikely and likely events? Decide on an interval to describe each, share and justify your answers.

Placing numbers on a number line

Consider rolling a 1 on a die out of 30 rolls. We determined that the likelihood of that event can be represented by the fraction 530. We can turn that fraction into a number between zero and one and plot it on the line by making it a decimal, or by dividing the numerator by the denominator.

5 ÷ 30 = 0.166

Where can we put this number on our probability line?

Press ‘Hint’ to reveal the solution.

0.166 would be plotted before the .25. Clearly it is not impossible, but it is pretty unlikely. If we multiply 0.166 by 100 (0.166 × 100 = 16.6), we know that there is a 16.6% chance of rolling a 1.

Creating number lines

If you would like, you can complete the next activity using TVO Mathify. You can also use your notebook or the following fillable and printable document.

For the following questions, determine the probability fraction, decimal, and percentage. Then, plot it on a probability line and describe the probability in words. You can also record a detailed written or audio recording about where each point would be on the probability line.

  1. You are spinning a wheel that is divided into eight equal sections. Three have a flower, one has a tree, one has a rock, one has a chair, one has a paper clip, and one has an eraser.
  1. What is the probability of landing on a section with a flower?
  2. What is the probability of landing on a natural item?
  1. There are 27 kids in an athletic camp program. 10 are on the basketball team, five are on the swim team, eight are on the soccer team, three are on the track team, and one participates in an individual sport. An athlete is picked at random to compete in a relay race.
  1. What is the probability of picking someone from the track team?
  2. What is the probability of picking someone on a sports team?
  3. What is the most likely team to be picked? Communicate your answers in percent’s decimals, fractions, and on a probability line.

Press the ‘TVO Mathify' button to access this interactive whiteboard and the ‘Activity’ button for your note-taking document. You will need a TVO Mathify login to access this resource.

TVO Mathify (Opens in new window) Activity (Open PDF in a new window)

Consolidation

Conduct a probability experiment

Choose 10 items from around you to place in an opaque bag or box or cover using another method. You should put doubles of some items. For example, in a bag there are: four coins, two socks, two pens, one eraser, and one bottle cap.

Determine the probability of each item in your bag in percents, decimals, fractions, words, and on a probability line.

Then, conduct an experiment with your bag and record the probabilities. Reach into the bag 20 times and record which item you retrieve from the bag. Return the items to the bag each time. Compare the results of your experiment to the probabilities recorded before the experiment.

  1. Are they the same? Are they different? Why?
  2. How could you change your bag to make one item have a probability of 80%?

Reflection

Reflect on and answer the following questions:

  1. Why is it important to represent probabilities in many different ways? How is it helpful?
  2. Which representation that you explored in this learning activity do you like the best? Why?

As you read through these descriptions, which sentence best describes how you are feeling about your understanding of this learning activity? Press the button that is beside this sentence.

I feel…

Now, record your ideas using a voice recorder, speech-to-text, or writing tool.

Press ‘Discover More’ to extend your skills.

Create a scenario where it would be necessary to determine a probability. Determine the probability in percents, decimals, fractions, words, and on a probability line.

Connect with a TVO Mathify tutor

Think of TVO Mathify as your own personalized math coach, here to support your learning at home. Press ‘TVO Mathify’ to connect with an Ontario Certified Teacher math tutor of your choice. You will need a TVO Mathify login to access this resource.

TVO Mathify (Opens in new window)