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Grade 8:  The Learning Equation Math

42.03 Independent Events

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Chance & Uncertainty

Refresher pp 112 - 113

Learning Outcomes:

The student will:

Theoretical and Experimental Probability for Coins

Theoretical probability is displayed in the opening screen of the virtual coin below. The green line shows the theoretical probability for heads and tails is 50%.

Experiment probability:  Toss a coin 20 times using the virtual coin provided by clicking the "?" button. The experimental probability does not have to match the theoretical probability. Click the "Reset" button if you wish to complete additional experimental probability trials.

The theoretical probability of the event is the fraction:

# ways the event can occur

total possible outcomes
 

The experimental probability for equally likely events is the fraction:

# favourable outcomes

# trials
 

In the Experiment, the theoretical probability of the event that heads comes up is:

P(H) =

1/2

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

In the Experiment, if heads comes up 13 of 20 times, the experimental probability of the event that heads comes up is:

P(H) = 13/20

= 0.65 = 65%

Theoretical Probability Practice

To be independent, one result cannot effect the probability of the other.

source:  http://www.targeon.org.uk/jrw/dat/quests.htm

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Note the following resources emphasize the total possible outcomes of two independent events. Theoretical probability:

# ways the event can occur
total possible outcomes

Tree Diagrams and Charts

Total Possible Outcomes

Total possible outcomes are required for theoretical probability calculations. A tree diagram (Method 1), table (Method 2) or list (Method 3) shows the possibilities. A tree diagram will include the possibilities and the branches, while the chart generally illustrates the same possibilities without the branches.  

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

 

Two Coins

Method 1

Complete a tree diagram to show all of the possible outcomes of flipping two coins.  Compare it to the tree diagram below.  It should include branches connecting the 2 possibilities for Coin 1 with the 2 possibilities for Coin 2. There are 4 possible outcomes when you flip two coins.

 

Another way of drawing the ttree diagram follows:

Event A:  Coin Flip

Event B:  Coin Flip

<

Htree.png (27212 bytes)

H
T
<

Ttree.png (27212 bytes)

H
T

number of possible outcomes = 2 x 2 = 4

Method 2

Make a chart to show the 4 possibilities for 2 coins. Compare your chart to the one given below.

  H T
H H, H H, T
T T, H T, T

Method 3

Make a list of all the possibile events (lists for 3 and 4 coins are included). Compare your list to the one given below.

#Coins

#Possibilities

List of possibilites

2

2 x 2 = 4

HH,  HT,  TH,  TT

 

 

Two Six-sided Dice

Method 1

The following shows a tree diagram of the possible sums when you toss two dice. To better understand the tree diagram, roll Die 1 until you get 1. Find the corresponding number in the chart below Die 1. Roll Die 2.  It will display the value from 1 to 6.   The branches beside each event in the left cell shows there are 6 possibilities for the second event. Each possibility for Die 1 connects with the 6 possibilities for Die 2.  There are 36 possibile outcomes when you roll two dice.

Event A: Die Toss

Event B: Die Toss

1tree.png (27212 bytes)

1
2
3
4
5
6

2tree.png (27212 bytes)

1
2
3
4
5
6

3tree.png (27212 bytes)

1
2
3
4
5
6

Event A: Die Toss

Event B: Die Toss

4tree.png (27212 bytes)

1
2
3
4
5
6

5tree.png (27212 bytes)

1
2
3
4
5
6

6tree.png (27212 bytes)

1
2
3
4
5
6

number of possible outcomes = 6 x 6 = 36

Method 2

Complete a chart to show the 36 possibilities for 2 six-sided dice thrown. Compare your chart to the one given below. 

This is the standard probability table. Notice that the possible outcomes of one die are placed across the top of the chart and the possible outcomes for the second die are placed down the left side of the chart. The sum of the two die are placed in the appropriate table cell. For example if the first and second die are both 1, then the sum of the dice is 2.

  1 2 3 4 5 6
1 2 3 4 5 6 7
2 3 4 5 6 7 8
3 4 5 6 7 8 9
4 5 6 7 8 9 10
5 6 7 8 9 10 11
6 7 8 9 10 11 12

Method 3

Make a list of all the possibile events. Compare your list to the one given below.

#Dice

#Possibilities

List of possibilites

2

6 x 6 = 36

11, 12, 13, 14, 15, 16
21, 22, 23, 24, 25, 26
31, 32, 33, 34, 35, 36
41, 42, 43, 44, 45, 46
51, 52, 53, 54, 55, 56
61, 62, 63, 64, 65, 66

 

The following animation shows the pattern of the possible sums when one tosses two. The 36 possible combinations are also displayed. This information is used in the next table to calculate the theoretical probability for the sum of two dice tosses.

 

Theoretical Probability for Sum of Two Dice Tossed

  1 2 3 4 5 6 7 8 9 10 11 12
Theoretical probability for the sum of two six-sided dice. 0 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36

 

 

Six-sided Die and Coin

Method 1

Complete a tree diagram to show all of the possible outcomes of flipping a coin and tossing a die.  Compare your tree diagram to the one given below. The table below shows the 6 possibilities for Die Toss followed by the 2 possibilities for Coin Flip. There are 12 possible outcomes when you roll a die and flip a coin.

Event A:  Die Toss

Event B:  Coin Flip

<

1tree.png (27212 bytes)

H
T
<

2tree.png (27212 bytes)

H
T
<

3tree.png (27212 bytes)

H
T
<

4tree.png (27212 bytes)

H
T
<

5tree.png (27212 bytes)

H
T
<

6tree.png (27212 bytes)

H
T

number of possible outcomes = 6 x 2 = 12

Method 2

Make a chart to show the 12 possibilities for a coin and  six-sided die tossed. Compare your chart to the one given below.

  H T
1 1, H 1, T
2 2, H 2, T
3 3, H 3, T
4 4, H 4, T
5 5, H 5, T
6 6, H 6, T

Method 3

Make a list of all the possibile events. Compare your list to the one given below.

#Dice

#Possibilities

List of possibilites

2

6 x 2 = 12

1H, 1T
2H, 2T
3H, 3T
4H, 4T
5H, 5T
6H, 6T

 

 

5 Sector Spinner and Coin

Complete a tree diagram to show all of the possible outcomes of flipping a coin and spinning a 5 sector spinner. Compare it to the tree diagram below. Branches connect the 5 Spinner possibilities with the 2 possibilities for Coin. There are 10 possible outcomes when you spin a 5 sector spinner and flip a coin.

Event A: Spin Wheel

Event B: Coin Flip

<

greentree.png (27212 bytes)

H
T
<

bluetree.png (27212 bytes)

H
T
<

whitetree.png (27212 bytes)

H
T
<

greytree.png (27212 bytes)

H
T
<

redtree.png (27212 bytes)

H
T

number of possible outcomes = 5 x 2 = 10

Remember when there are two independent events, the number of possible outcomes equals the number of possible outcomes of the first event times the possible outcomes of the second event.

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P(Event 1,Event 2) = P(Event 1) x P(Event 2)

Probability of Two Independent Events Calculation

 

Independent Events: Probability of Tossing Die and Spinning a Colour

Event A:  Toss Die

Event B:  Spin Wheel

 

Toss one six-sided dice and spin a five sector color spinner.   What is the theoretical probability of a particular number and color being selected?

Solution:

You can use a real die and spinner or use the virtual ones above.  The theoretical probability of of any number being tossed is 1/6, while the theoretical probability of the spinner stopping on each color is 1/5. Examples:

  • P(5, green) means the probability of tossing 5 and spinning green

    P(5, green) = P(5) x P(green) = 1/6 x 1/5 = 1/30 or 3.333...%

  • P(3, red) means the probability of tossing 3 and spinning red

    P(3, red) = P(3) x P(red) = 1/6 x 1/5 = 1/30 or 3.333...%

Complete 30 trials and complete the experimental probability.

Two Events Theoretical Probability Experimental Probability
P(5, green) = 1/6 x 1/5 = 1/30 = 0.0333...approximately3.3% /30
P(3, blue) = 1/6 x 1/5 = 1/30 = 0.0333...approximately3.3% /30

 

Independent Events: Probability of Rolling Dice and Spinning a Colour

Event A:  Toss Dice

Event B:  Spin Wheel

Toss two six-sided die and spin a five sector color spinner.   What is the probability of a particular number and color being selected?

Solution:

You can use real dice and spinner or use the virt ual ones.  The probability of of any number being tossed varies, while the probability of the spinner stopping on each color is 1/5.  Examples:

  • P(5, green) means the probability of tossing 5 and spinning green

    P(5, green) = P(5) x P(green) = 4/36 x 1/5 = 4/180 or 2.222...%

  • P(3, red) means the probability of tossing 3 and spinning red

    P(3, red) = P(3) x P(red) = 2/36 x 1/5 = 2/180 or 1.111...%

Complete 180 trials. Record the number of times the selected two events occur in the experiment, then complete the experimental probability. Will the theoretical and experimental probability always be the same? answer: No

Two Events Theoretical Probability Experimental Probability
P(5, green) = 4/36 x 1/5 = 4/180 = 0.0222...approximately2.2% /180
P(3, red) = 2/36 x 1/5 = 2/180 = 0.0111...approximately1.1% /180

 

 

Independent Events: Probability of Drawing Two Cards

 

To be independent, you must put the first card back, before drawing the second.  This is equivalent to drawing from two different decks of cards.

Complete the appropriate number of  trials needed for the experimental probability. Will the theoretical and experimental probability always be the same? answer: No

Two Cards selected Theoretical Probability Experimental Probability
P(©,¨) = 1/4 x 1/4

1/16 = 6.25%

/16

 

Complete the appropriate number of  trials needed for the experimental probability. Will the theoretical and experimental probability always be the same? answer: No

Two Cards selected Theoretical Probability Experimental Probability
P(4,7) = 1/13 x 1/13

= 1/169approximately0.05%

/169

 

 

More Independent Events

Linked Source

Toss 6-sided die, Flip Coin, Spin 5 Sector Spinner, Draw a Card

Theoretical probability is displayed in the opening screen of each probability explorer selected. The green line shows the theoretical probability for each event.

Experiment probability:  Clicking the "?" button for Event A and Event B starts a probability experiment. The experimental probability will likely approach the theoretical probability for a large sample. Click the "Reset" button for Event A and Event B if you wish to complete additional experimental probability trials.

Note: the sectors for the spinners included are the same size. Do not assume that other resources will do the same. If you are using a new spinner take a few moments to examine the sector size.

 

Event A

Event B

 

Event A

P(A) Event B P(B) Theoretical Probability

P(A,B) = P(A) x P(B)

Toss 3
(6-sided die)

1/6

Toss 4
(6-sided die)

1/6 1/6 x 1/6approximately0.028approximately2.8%

Toss 2
(6-sided die)

1/6

Toss 5
(6-sided die)

1/6 1/6 x 1/6approximately0.028approximately2.8%

Toss 6
(6-sided die)

1/6 Flip Tails 1/2 1/6 x 1/2approximately0.083approximately8.3%
Flip heads 1/2

Toss 3
(6-sided die)

1/6 1/2 x 1/6approximately0.083approximately8.3%
Flip tails 1/2 Spin green 1/5 1/2 x 1/5 = 0.1 = 0.10%
Draw club 1/4 Spin yellow 1/5 1/5 x 1/4 = 0.05 = 5%
Draw 6 1/13 Flip heads 1/2 1/13 x 1/2approximately0.038approximately3.8%
Draw heart 1/4 Flip heads 1/2 1/4 x 1/2 = 0.125 = 12.5%

 

More challenging tasks:

Toss even #
(6-sided die)

3/6 spin blue or red on 6 sector spinner 2/6 1/2 x 1/3 = 0.1666...approximately16.6%

Toss odd #
(6-sided die)

3/6 spin yellow, blue or red on 6 sector spinner 3/6 1/2 x 1/2 = 0.25 = 25.0%

Toss # less than 5
(6-sided die)

4/6

Toss # greater than 4
(6-sided die)

2/6 2/3 x 1/3 = 0.222....approximately22.2%
An ace or king 8/52

Toss # that is a multiple of 3
(6-sided die)

2/6 4/26 x 1/3approximately0.0513approximately5.1%
Club or diamond 26/52

Toss # that is a multiple of 2
(6-sided die)

3/6 1/2 x 1/2 = 0.25 = 25.0%
Face card 12/52

Toss # that is greater than 2
(6-sided die)

4/6 3/13 x 2/3approximately0.1538approximately15.4%

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Review:

Lesson on Probability of Independent Events

E-Lab: Exploring Independent Events - choose 2 of the 3 options

 

Enrichment:

Two Events: Addition (‘Or’) Law

Two Events: Multiplication (‘And’) Law

 

Key Terms:

expected probability, independent events, dependent events, composite numbers, difference, lowest terms, prime number

 

Prerequisite Skills:

Using Probability: Grade 8 Lesson 42.02

Using Probability: Grade 8 Lesson 42.02

Probability: Grade 7 Lesson 42.02

Exploring Probability:  Grade 7 Lesson 42.01

Adding Fractions with Unlike Denominators:  Grade 8 Lesson 12.02

Subtracting Fractions (Unlike Denominators):  Grade 8 Lesson 12.03

Multiplying Fractions:   Grade 8 Lesson 12.04

Dividing Fractions:  Grade 8 Lesson 12.05

 

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Comments to:  Jim Reed

Started September, 1998. Copyright © 1999, 2000