Minds On
Theoretical and experimental probability
In the following video, Teacher Sarah reviews theoretical and experimental probability.
True or false?
Consider the following statements. For each, select True or False, then press ‘Check Answer’ to see how you did.
Action
Review: Important definitions
For each term, select the correct definition.
Exploring independent events
Scenario 1: Flipping two coins
To win this game, the player must flip two coins and get “heads” on both.
When there are two coins, there are two independent variables. Each coin has its own outcome. Each coin has a , 50%, or 0.5 chance of landing on “heads.” In order to have the outcome of both coins landing on “heads,” we need to combine the theoretical probability using multiplication.
Press ‘Formula’ to review how to determine theoretical probability.
Theoretical probability
Theoretical probability of getting two “heads”
When a single coin is flipped, each flip has a ½, 50%, or 0.5 chance of landing on “heads.”
What is the theoretical probability of flipping two coins and having both of them land on “heads”? Explore the following table.
| Coin 1 | Coin 2 | Combined probability | |
|---|---|---|---|
| Chance of getting “Heads” | or 50% | or 50% | or 25% |
| Calculating the combined probability: To find the theoretical probability of getting “heads” on both coins, we multiply the two probabilities together: | |||
Scenario 2: Flipping over cards
Three classmates are playing a game that involves two identical piles of 40 cards. In each pile, half of the cards have “Heads” written on them, and half have “Tails” written on them.
To take a turn, a player flips over the top card of each pile.
Each time a player takes a turn, there are three possible results:
After each turn, the cards that have been flipped are put back into the piles. The events are therefore independent.
How the game is played
The player who scores the most points wins.
- Player A scores a point each time they get Heads/Heads.
- Player B scores a point each time they get Heads/Tails.
- Player C scores a point each time get Tails/Tails.
Brainstorm
Make your predictions!
Throughout this learning activity, you can record your answers digitally, orally, or in print.
- What is theoretical probability for each of the possible outcomes? (Note that each time a card is flipped, there is an equal chance of getting “heads” or “tails.”)
- Is this game fair? Do all players have an equal chance of winning?
Experimental results
The game is played for 20 turns, and the data is recorded in the following table:
| Heads-Heads | Heads-Tails | Tails-Tails | |
|---|---|---|---|
| Turn 1 | X | ||
| Turn 2 | X | ||
| Turn 3 | X | ||
| Turn 4 | X | ||
| Turn 5 | X | ||
| Turn 6 | X | ||
| Turn 7 | X | ||
| Turn 8 | X | ||
| Turn 9 | X | ||
| Turn 10 | X | ||
| Turn 11 | X | ||
| Turn 12 | X | ||
| Turn 13 | X | ||
| Turn 14 | X | ||
| Turn 15 | X | ||
| Turn 16 | X | ||
| Turn 17 | X | ||
| Turn 18 | X | ||
| Turn 19 | X | ||
| Turn 20 | X |
Test Your Skills!
Analyzing the results
Record your observations using the method of your choice.
- Based on the game results, determine the experimental probability of winning for each of the players.
- Compare the theoretical probability to the experimental probability.
- What do you notice?
- What will happen if the players play another 20 rounds of the game?
Note to teachers: See your teacher guide for collaboration tools, ideas and suggestions.
Exploring dependent events
Scenario: The grab bag
Imagine a bag that contains 16 items:
- four pencils
- four pens
- four rulers
- four erasers
To take a turn, a player draws a single item from the bag.
Each time a player draws an item from the bag, this item is set aside (not put back in the bag). What happens on the first turn has a direct impact on what happens on the second turn. They are, therefore, dependent events.
Brainstorm
Make your predictions!
Working independently or with a partner, consider the following questions. Use the method of your choice to record your answers.
- What is the probability of drawing a pencil from the bag containing 16 items?
- Imagine that in the first turn, a ruler is drawn from the bag and set aside.
- How many items are now in the bag?
- What is the new theoretical probability of drawing a pencil from the bag?
Experimental results
The players took 16 turns, recording the items drawn in the following table. Reminder: After each turn, the item was set aside.
Complete the Data Table: Grab Bag in your notebook or use the following fillable and printable document. If you prefer, use another method to record your ideas.
| Turn | Items Selected | Total items in bag | Pencils remaining in bag | Theoretical probability of drawing a pencil |
|---|---|---|---|---|
| 0 | NONE | 16 | 4 | or or 25% |
| 1 | Ruler | 15 | 4 | |
| 2 | Pen | 14 | ||
| 3 | Eraser | |||
| 4 | Eraser | |||
| 5 | Pen | |||
| 6 | Pencil | |||
| 7 | Eraser | |||
| 8 | Ruler | |||
| 9 | Pen | |||
| 10 | Eraser | |||
| 11 | Pencil | |||
| 12 | Pen | |||
| 13 | Ruler | |||
| 14 | Pencil | |||
| 15 | Ruler | |||
| 16 | Pencil |
Press the ‘Activity’ button to access the Data Table: Grab Bag.
Test Your Skills
Analyzing the results
Record your observations using the method of your choice.
- Compare the experimental results to the theoretical probability of selecting a pencil.
- What do you notice about the experimental probability each turn?
- Do the experimental results match the theoretical probability?
- How would replacing the item in the bag after each turn change the theoretical probability?
Consolidation
Carnival games
Fairs and carnivals offer lots of entertainment and activities. For example, there are many carnival games inviting you to try your luck and win a prize.
Probability can help you decide how likely it is that you will have a positive outcome in any given game.
Duck pond game
For this game, there is a bucket of water containing 20 rubber ducks. There is also a circular spinner divided into six segments, one of which has a picture of a duck on it. When the player presses a button, the spinner spins and lands on one of the six segments.
How the game is played
To play the duck pond game, a player completes tasks. Press on each task to learn more.
The player draws one rubber duck from the bucket containing 20. Half of the ducks have “prize duck” written on the bottom. If the player draws a “prize duck,” they receive a small prize.
The player spins the six-section spinner. If the spinner lands on the “duck” section, the player receives a medium prize.
If the player wins a small prize and a medium prize, they can trade them for a large prize.
Brainstorm
Thinking about the duck pond game
- Are the first two tasks independent or dependent events?
- What is the theoretical probability of winning a small prize at the duck pond?
- What is the theoretical probability of winning a medium prize by spinning the spinner?
- Combine these two probabilities using multiplication. What is the theoretical probability of the player being able to trade in the small and medium prizes for a large prize?
Reflection
As you read the following descriptions, select the one that best describes your current understanding of the learning in this activity. Press the corresponding button once you have made your choice.
I feel…
Now, expand on your ideas by recording your thoughts using a voice recorder, speech-to-text, or writing tool.
When you review your notes on this learning activity later, reflect on whether you would select a different description based on your further review of the material in this learning activity.
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