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Grade 7: The Learning Equation Math

42.01 Exploring Probability

 Chance & Uncertainty Refresher pp 88-89

Learning Outcomes:

The student will:

## Tree Diagrams and Charts

### Total Possible Outcomes

Total possible outcomes are required for theoretical probability calculations. A table shows the possibilities.

Explore the total possible outcomes of the following pairs of events.

### One Coin

Complete a chart to show all of the possible outcomes of flipping one coin.  Compare it to the table below.

### Event:  Coin Flip

H
T

number of possible outcomes = 2

### One Six-sided Die

The following shows the possible outcomes when you toss a die. Roll the die several times until you see the six different numbers displayed on the die

### Event: Die Toss

1

2

3

4

5

6

number of possible outcomes = 6

## Theoretical Probability of a Single Event

The theoretical probability of the event is the fraction:

# ways the event can occur
total possible outcomes

### Coin Flip

When you flip a coin, there are only 2 sides - heads and tails. The theoretical probability for each is:

P(H) = 1/2 = 0.5
P(T) = 1/2 = 0.5

### Spin Wheel

When a spinner has 5 equal sized sectors of green, red, yellow, and blue, the theoretical probability for each is:

P(red) = 1/5 = 0.20
P(orange) = 1/5 = 0.20
P(yellow) = 1/5 = 0.20
P(green) = 1/5 = 0.20
P(blue) = 1/5 = 0.20

### Toss Die

When you toss a 6-sided dice, there are 6 sides. The theoretical probability (to the nearest hundredth) for each is:

P(1) = 1/6 = 0.17
P(2) = 1/6 = 0.17
P(3) = 1/6 = 0.17
P(4) = 1/6 = 0.17
P(5) = 1/6 = 0.17
P(6) = 1/6 = 0.17

Theoretical probability does not change. In the first examples each individual event had the same probability. The next example shows events with different probabilities.

## Experimental Probability

### Coin Flip

The experimental probability for equally likely events is the fraction:

# favourable outcomes
total outcomes

If you toss a coin 50 times and you end up with 20 heads and 30 tails, the experimental probability is:

P(H) = 20/50 = 0.4

P(T) = 30/50 = 0.6

Theoretical experiment changes from experiment to experiment. If the experiment is fair, the theoretical and experimental probability should be very similar if a large number of trials are made.  Try flipping a coin yourself.  Flip it 50 times and record the numbers of heads and tails.  Divide your totals by 50 to get the experimental probability.  Coin toss simulations are available on the internet.   Try these:

Coin Toss

Theoretical and Experimental Probability for Coins

Toss a coin 20 times. If heads comes up 13 times, then the favorable outcomes of the event that heads comes up is 13.

### Coin Flip

The theoretical probability of the event is the fraction:

# ways the event can occur
total possible outcomes
Referring to the situation above, the theoretical probability of the event that heads comes up is:

P(H) = 1/2 = 0.5 = 50%

The experimental probability for equally likely events is the fraction:

# favourable outcomes
total outcomes
Referring to the situation above, the experimental probability of the event that heads comes up is:

P(H) = 13/20 = 0.65 = 65%

## Theoretical and Experimental Probability for Coin Tosses

Click on one of the toss buttons to select how many times you want the coin tossed. If you select the consecutive series option, the program will work through all 6 samples starting with single toss ending with 100 000 tosses.

The theoretical probability for both heads and tails is 0.5 or 50%. The experimental probability for a single toss is 1.0 pr 100% (it has to be either heads or tails). Compare the theoretical and experimental probability results for each toss option as you complete the following exercise:

 TURNS HEADS TAILS #heads Probablility Probablility #tails Experimental P(H) Theoretical P(H) Experimental P(T) Theoretical P(T) 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 This will complete all 6 options consecutively.

Source and permission for noncommercial use: Monte Carlo Introduction

## Rolling Dice Probability

For equally likely events the theoretical probablility is:

# ways the event can occur
total possible outcomes
• The probability for 4, 5 and 6-sided dice are displayed below:

### Theoretical Probability

The number of faces on dice can vary.  If there is a different number on each face, the theoretical probability will be:

1
#faces
# tossed 4 Faces 5 Faces 6 Faces
1 P(1) = 1/4 P(1) = 1/5 P(1) = 1/6
2 P(2) = 1/4 P(2) = 1/5 P(2) = 1/6
3 P(3) = 1/4 P(3) = 1/5 P(3) = 1/6
4 P(4) = 1/4 P(4) = 1/5 P(4) = 1/6
5 P(5) = 1/5 P(5) = 1/6
6 P(6) = 1/6

• Roll the die 6 times. Did the experimental probability match the theoretical probability of 1/6 (17%) for each number?
• Click reset.  Roll the die 120 times.  Did the experimental probability match the theoretical probability of 1/6 (17%) for each number?

INTRODUCTION TO PROBABILITY

Review:

## Spinners

Exploring Probability

Try the spinner for to the experimental probability investigations.

Lesson on Introduction to Probability - contains cool dice roll

Enrichment:

Percent and Probability

Introduction to Probability

Seth H. Jackson's Assignment 6 - dice roll applet

Lesson on Sample Spaces - great, but sample space not part of program vocabulary

Lesson on the Complement of an Event - great, but compliment not part of program vocabulary

Central Limit Theorem Demo Applet

 Key Terms: A-E F-J K-O P-R S-Z

probability, outcome, experimental probability, theoretical probability, frequency, independent events, sectors, central angle

Prerequisite Skills:

Included in this lesson.

Included in this lesson.

Included in this lesson.

Included in this lesson.

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