Lesson 2

Lesson 2: Theoretical Probability

 

Focus

 

When planning a summertime camping trip, people check the weather forecast for the possibility of favourable conditions. In Alberta a common saying is, “If you don’t like the weather, just wait a minute.” This saying indicates that the weather in Alberta can be unsettled at times.

 

A weather forecast tries to predict the likely outcome of the weather, but sometimes it may be wrong. Why might this be?

 

Lesson Outcomes

 

At the end of this lesson you will be able to

• determine the theoretical probability of an event
• distinguish between experimental and theoretical probability

Lesson Questions

 

In this lesson you will investigate the following questions:

• How does theoretical probability differ from experimental probability?
• What is the theoretical probability of an event?

Assessment

 

Your assessment may be based on a combination of the following tasks:

• completion of the Lesson 2 Assignment (Download the Lesson 2 Assignment and save it in your course folder now.)
• course folder submissions from Try This and Share activities
• additions to Glossary Terms
• work under Project Connection

Materials and Equipment

 

You will need

• calculator
• two standard six-sided dice (virtual dice are also available in the lesson)
• assortment of coins
• deck of cards

Discover

 

In Lesson 1 of this module you experimented with experimental probability by rolling a six-sided standard die multiple times and recording the results. The experimental probability was determined based on the results of the experiment.

 

For example, if you rolled a four 10 times in 36 rolls of the die, then your experimental probability would have been Expressed as a percent, the probability of rolling a four in your experiment was approximately 27.8%. Does this sound reasonable? Would you use your experimental results to help you make a good decision? That is, if you were asked to predict the outcome of the next roll of the die, could you confidently say that it would be a four?

 

Now dive deeper and continue to explore your predicting abilities. This time it will involve calculating the absolute difference when you roll two dice!

 

Try This 1

 

Open Dice Roller and use it to complete the experiment and answer the following questions.

 

In this experiment you will be rolling two dice and recording the difference in their outcomes (you will always subtract the smaller value from the larger value).

 

1. Prior to rolling the dice, list the set of possible outcomes for the experiment. Keep in mind that your possible outcomes are based on the difference between the values on the two dice.
2. Next, make predictions to the following questions.
   1. Will some of the outcomes occur more frequently than others?
   2. Which outcome(s) do you expect will occur most often?
   3. Which outcome(s) do you expect will occur least often?
3. Roll the dice a total of 36 times. Using the difference between the value on the two dice as the outcome, record the tally and frequency in a table similar to the one shown.
   
   

   Outcome	Tally	Frequency	Experimental Probability
   0
   1
   2
   3
   4
   5
   Total	36	36

4. Next, determine the experimental probability for each outcome. Record the results in the last column of your table.

5. Based on the results of your experiment, answer the following questions.

   1. Were your predictions of the least and the most expected outcomes correct?
   2. How confident are you that your experimental probability results are very close to those of other students or groups performing the same experiment?
   3. If you were to perform this experiment with 10 times the number of dice rolls, how might this change the experimental probability? If you had 100 times the number of dice rolls, would the experimental probability change? Explain.

Save your responses in your course folder. You will need these predictions for comparison after the experiment.

 

Share 1

 

With a partner or in a group, discuss the results of your experiment and your responses to the questions in Try This 1. Compare your findings. Explain why there may be similarities and differences.

 

Save a summary of your discussion in your course folder.

In Try This 1 you listed all possible outcomes of the experiment. Listing all the possibilities is also known as determining the sample space.

 

Try This 2

1. Copy and complete the following sample space table to show all possible differences when rolling two dice. The first row and column of outcomes have been completed for you.
   
   

   |Difference|		Outcome of Die 1
   		1	2	3	4	5	6
   Outcome   of   Die 2	1	0	1	2	3	4	5
   	2	1
   	3	2
   	4	3
   	5	4
   	6	5

2. What is the total number of outcomes in the sample space table?
3. Using the sample space table, predict the probability of rolling a difference of 0, 1, 2, 3, 4, and 5.
4. Copy and summarize your findings in a table like the following. The frequency table probabilities are from Try This 1, and the sample space table probability is from Try This 2 question 1.
   
   

   Outcome	Frequency Table Probability (from Try This 1)	Sample Space Table Probability (from Try This 2 question 1)
   0
   1
   2
   3
   4
   5

5. Explain how the frequency table and the sample space table are similar.
6. Explain how the frequency table and the sample space table are different.
7. Do either of these methods of representation—frequency table or sample space table—better represent the probability of rolling a particular difference? Explain.

Save your work in your course folder.

Explore

 

In Discover you constructed a sample space table of all possible outcomes. This table can be used to determine the theoretical probability of the different outcomes.

 

Theoretical probability is determined by calculating the number of favourable outcomes and the total number of possible outcomes, whereas experimental probability is determined by performing an experiment, as you did in Lesson 1 of this module.

 

The formula for calculating theoretical probability is the same as experimental probability.

 

Source: MathWorks 12 Student Book/Teacher Guide. (Vancouver:Pacific Educational Press, 2011.)

 

	Experimental Probability	Theoretical Probability
Definition	the ratio of the number of occurrences of a particular event during an experiment or trial to the total number of trials conducted	the ratio of one or more events of interest to the total number of possible outcomes
Formula

 

For example, when flipping a single coin, the result will be either heads or tails. There are only two possible outcomes.

 

The theoretical probability of flipping a head is

 

In Try This 3 you will further explore how theoretical probability and experimental probability compare.

 

Try This 3

 

Predict how many heads will appear when 16 coins are flipped. Does your prediction change if there are 32, 48, or 64 coins flipped? Test your prediction with an experiment.

 

Open Coin Flipper and click on 16 for the number of coins. Then select “Flip Coins.” Note that this experiment can also be completed using a real coin by simply flipping it 16 times, and recording the results. You can also flip 16 coins at once.

 

coin images: Royal Canadian Mint. All Rights Reserved.

 

Predict the number of heads you expect for each of the next four trials of 16 coins, and record your predictions in a table similar to the one that follows.

 

	Cumulative Number of Coins Flipped	Predicted Number of Heads	Actual Number of Heads
1st set of 16 coins
2nd set of 16 coins
3rd set of 16 coins
4th set of 16 coins
5th set of 16 coins
Total

 

Flip the coins four more times, for a total of five 16-coin trials.

 

Record in the table the total number of heads that appear.

 

Save your responses in your course folder.

 

Share 2

 

With a partner or in a group, discuss your findings from Try This 3. Then use your results to answer the following questions.

1. Combine your results. Determine the total number of flips of the coin. Determine the total number of heads that appeared. What is the experimental probability of flipping a head based on the combined data?
2. Calculate the theoretical probability of flipping a head. Are the two probabilities close?
3. How do you think these numbers would compare if you flipped 10 times as many coins?

If you wish, find more people to add to your group and combine the results. Are your findings closer to the theoretical probability?

 

Save your responses in your course folder.

In Try This 3 you may have noticed that the theoretical and experimental probabilities were similar. Also, the experimental probability came closer to the theoretical probability as you did more trials. In the next activity you will determine some theoretical probabilities for events more complex than flipping coins.

 

Let’s experiment with a deck of cards.

 

Try This 4

 

 

A standard deck of playing cards consists of 52 cards labelled as follows:

• Four suits include spades, hearts, diamonds, and clubs.
• Two suits are red: hearts and diamonds.
• Two suits are black: spades and clubs.
• In each suit there are 13 members: ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, and king.
• Jacks, queens, and kings are called face cards.

If you were to randomly draw a card from a shuffled deck, find the probability for each of the following questions.

1. What is the probability that the card would be a 2 of hearts?
2. What is the probability that the card would be a spade?
3. What is the probability that the card would be a face card?
4. What is the probability that the card would be a black 9?
5. Suppose you drew a card and looked at it. You then returned the card to the deck, shuffled the deck, and repeated the process until you had looked at 20 cards. How many times would you expect to see the following?
   1. a red card
   2. a face card
   3. a 7 of clubs

Save your responses in your course folder.

Self-Check 1

After completing Try This 5, you should have noticed that when an item or object is not replaced after being picked, the sample space or total number of possible outcomes will change. Check your knowledge of this in Self-Check 2.

 

Self-Check 2

Self-Check 3

 

You have been offered the opportunity to complete an online customer satisfaction survey for Home Renos Supply Depot. All survey entries receive a prize. After submitting your comments, you are entered into a draw to determine your prize. The grand prize is an all-inclusive one-week paid trip to Hawaii. Ten thousand (10 000) surveys are entered into the draw, and the prizes are distributed as follows:

 

Prize	Number of Prizes of This Type
5% off your next purchase	7922
10% off your next purchase over $250	1577
30% off your next purchase over $500	500
All-inclusive one-week paid trip for two to Hawaii	1

 

Elia and Erryn have filled in satisfaction surveys and are discussing the probability of winning the grand prize. Elia says she thinks she has a 1 in 4 chance of going to Hawaii because there are four prizes available and she will win one of them. Erryn says that sounds too good to be true.

1. Which girl do you agree with? Explain. Answer
2. What is the formula for the probability of winning a certain prize? Answer
3. What is the probability of winning a prize of 30% off your next purchase over $500? Answer
4. What is the probability of winning a prize other than 5% off your next purchase? Answer
5. If you checked the prizes of 500 people, how many do you expect received the 5% discount? Answer

---

In Self-Check 3 you may have noticed that, in fact, there is not a 1 in 4 chance of winning the trip to Hawaii. Why? Because the four options do not have equal probabilities.

 

The probability of the individual events must be equal to use the formula

 

You also saw that the probability of an event that must happen is 1, and the probability of an event that cannot happen is 0.

 

The diagrams illustrate these ideas.

 

 

 

Self-Check 4

 

You have a bag with 9 red marbles, 6 blue marbles, and 5 green marbles. A marble is selected and then put back in the bag. Answer each of the following questions as a fraction in lowest terms, as a decimal, and as a percentage. What is the probability of randomly selecting the following colours of marbles?

1. a green marble Answer
2. a red or blue marble Answer
3. a black marble Answer
4. a non-green marble Answer
5. a pink marble Answer
6. Explain why the probability of selecting a red marble is not Answer

Self-Check 5

Connect

 

Lesson 2 Assignment

Lesson 2 Summary

 

In this lesson you looked at theoretical probability and how it relates to experimental probability, as seen in Discover when you used the absolute value of the difference of rolling two dice.

 

Theoretical probability explains what should happen in an experiment, and experimental probability explains what actually happened when the experiment was performed.

 

You also learned that as the number of trials in an experiment increases, the experimental probability approaches the same value as the theoretical probability.

 

The formula for experimental and theoretical probability is the same.

 

 

In Lesson 3 you will begin to explore odds. Odds provide an alternate way to express the chance of something occurring.