Minds On
Exploring probability
In the following video, Sarah reviews theoretical and experimental probability.
As you explore, consider the following statements and determine whether they would be true or false. Throughout this learning activity, you can record your ideas digitally, orally, or in print.
- The probability of landing on tails when you flip a coin is 65%.
- The probability of pulling a five randomly from a deck of 52 cards is 5/100.
Action
Theoretical vs. experimental probability
Theoretical probability is the chance that an event will occur in theory.
As a fraction, the number of favourable outcomes is the numerator and the possible outcomes in the denominator.
For example, when you flip a coin there are two possible outcomes. Two becomes the denominator.
Outcome 1: you will flip heads.
Outcome 2: you will flip tails.
If you’re determining the probability of heads being flipped, there is one favourable outcome. 1 is the numerator. Record the probability. .
This probability can also be rewritten as a decimal (0.5) or as a percent (50%) to express the chance that the coin flipped will be heads.
Experimental probability is measuring the likelihood of an event based on performing an experiment. The denominator represents the number of possible outcomes, just like in theoretical probability.
If you flip a coin 10 times there are 10 ways for it to land. 10 will be the denominator.
If you actually conduct this experiment, there are many possibilities. The numerator will be the number of times the favourable outcome occurs, or the number of times my coin land on heads. If the coin landed on heads six times, the numerator will be 6.
The fraction will be .
The probability can also be represented as a decimal (0.6) or a percent (60%). In this case, the experimental probability would be different than the theoretical probability.
However, that will not always be the case. There may be times where you flip the coin 10 times and land on each side an equal number of times! As you increase the amount of possible outcomes or the amount of times you conduct the experiment, you will get closer to the theoretical probability.
Complete the Sorting Chart in your notebook or use the following fillable and printable document. Sort each example into the chart to determine whether it represents a theoretical or experimental probability.
Press the ‘Activity’ button to access Sorting Chart.
Outcomes
An outcome is the possibility of an event occurring, or of the result of an experiment.
Teacher Troy will conduct some experiments to help our understanding of probability and possible outcomes.
Exploring Scenarios
Explore the following scenario and give advice to the player:
A group of students are playing a board game. Player A is about to have their turn. They are really disappointed because they want to roll a seven, but it has already rolled the last two turns. They think that there’s no way it will happen three times in a row! The player shakes the two dice for 10 whole seconds and rolls.
Using what you know about probability, what could you tell player A? Should the player be disappointed that a seven has been rolled two times already? Is it likely for them to get their desired outcome?
Record your ideas in a method of your choice and be ready to share your thoughts.
Press the ‘Hint’ button to check your answers.
In this case, the favourable outcome is rolling a 7.
In determining both theoretical and experimental probabilities, it is important to understand what the possible outcomes of a scenario are.
For independent events, the outcome of one event does not impact the outcome of another event. For example, every time you roll a die you have an equal chance of rolling each number. In the scenario above, it doesn’t matter that a 7 was rolled twice before. The possibility of each outcome will not change.
Record your responses to the following questions in a method of your choice.
- Can you think of other examples where the outcome of one event will not impact the outcome of another event?
- Can you think of examples where this isn’t true?
Tree diagrams
Consider the probability of two independent events, such as rolling two sixes in a row on a die. The probability of one event does not impact the probability of the other event.
To determine the probability of rolling two sixes in a row, you need to determine all of the possible outcomes for the order of two rolls.
Determine all the possibilities for roll #1. Roll #1 can be any number from one to six.
A tree diagram is used to record all the possible outcomes for roll #1. The outcomes are listed on different lines extending from the center point. The lines are labelled with the numbers 1, 2, 3, 4, 5, 6.
Then, determine all of the possibilities for roll #2 following each of the possibilities of roll one. Roll #1 can be any number from one to six. Roll #2 after each of those six rolls, can also be any number from one to six.
The tree diagram now extends to show the possible results of the second roll. The first set of outcomes are listed, and beside each outcome extends another set of six lines to indicate the six outcomes. For the first roll of 1, the next roll can also have 1, 2, 3, 4, 5, 6 as the possible outcomes. For the first roll of 2, the next roll can also have 1, 2, 3, 4, 5, 6 as the possible outcomes. This is true for 3, 4, 5, 6 as the first roll.
Then, list all possible outcomes. Record as: (first roll, second roll). Roll One is listed first in the bracket followed by a comma, and then roll two is listed. Record you answers in a method of your choice.
Press ‘Hint’ to reveal an example of how to record your outcomes.
Hint(1,1), (1,2), (1,3), (1,4), (1,5), (1,6)
How many outcomes are possible? With the total possible outcomes as the denominator, determine the probability of rolling two sixes in a row. Add the number of times a (6, 6) can occur as the numerator.
Build your own tree diagram
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.
Press the ‘TVO Mathify’ button to access this resource and the ‘Activity’ button for your note-taking document.
TVO Mathify (Opens in new window)Activity(Open PDF in a new window)Consolidation
Show what you know
Experimental vs. theoretical probability
Create your own experiment using objects that are available to you to determine the probability of two independent events. You may use a die, cards, spinners, coins, or a bag of different items or a digital program.
Answer the following questions:
- What are the independent events?
- How many possible outcomes are there?
- What is the experimental probability of picking each item twice in a row?
- Choose one of the possible outcomes to be your desired outcome. What is the theoretical probability?
- Conduct your experiment. What was the experimental probability of your desired outcome? How is it different than the theoretical probability?
Reflection
Reflect on and answer the following questions.
- Why is it important to understand the difference between experimental and theoretical probabilities?
- How does conducting experiments add to your understanding of probability?
How do you feel about what you have learned in this activity? Which of the next 4 sentences best matches how you are feeling about your learning? Press the button that is beside this sentence.
I feel…
Now, record your ideas about your feelings using a voice recorder, speech-to-text, or writing tool.
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)