The Fundamental Counting Principle
Independent Events
Probabilities that we normally use, involve more than 1 event. Such as rolling 2 dice, drawing 5 cards,
choosing a 2 or 3 piece clothing ensemble, making meal choices of appetizer, entrée, and dessert,
Choosing the order of classes to take next year, guessing a forgotten locker combination, etc.
Think
About it.
Learn
about it.
1
Apply
your
knowledg
e.
FCPTD
Examples of how the fundamental counting principle can be used can be f ound here.
HERE
Follow this link to learn about the sample space of two independent events can be drawn in a rectangular array
f or a coin and a single die. Click the ">" one time to see the sample space and the probability of these 2 events.
IACE
Now that y ou'v e seen what the sample of a sample space in two dif f erent f ormats, a tree diagram and a table,
Y ou are going to construct a sample space f or 2 Independent Events, a coin and a spinner in both a tree diagram and a table.
Click on the "Coin and Spinner" tab at the bottom of this worksheet.
When y ou are f inished with y our table, y our tree diagram and a f ew questions, Click the Lesson 4 Tab to come back here.
2
3
Experiment
Demonstration of the f undamental counting principle using a TREE DIAGRAM sample space
4
When rolling 2 dice, y ou must consider them 2 independent events. Each die has 6 sides.
Use the Counting Principle to f ind the total number of outcomes f or 2 dice.
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( If y ou f orgot what the counting principle is about, go back up to the 3 links in the prev ious section of this page)
(Or, if y ou want to see the sample space f or tossing two dice, try this Sample Space Link )
Now go to the "2 Dice" tab at the bottom of this page to construct the sample space f or rolling 2 dice.
Click the Lesson 4 Tab to come back here.
Now y ou're ready to experiment with Independent events. Y ou'll be going to the v irtual dice toss page.
Y ou can use this experiment to see what y our "2 Dice" worksheet should look like.
Keep track of the bar graph on the right side of the screen. Both Experimental and Theoretical probability are display ed.
Be sure to set the Number of Dice to 2.
Lets go there now ->
Virtual Dice Toss
Each die roll independently of the other.
For example P( 2 on the second die) does not depend on the result of tossing the f irst.
Use the sample space on the "2 Dice" worksheet if necessary , to help answer these questions.
Think
about it.
1st Die
5
P(6) =
2nd Die
1
6
P(6) =
Both Dice
1
6
P(double 6) =
1
36
Check this answer by using y our sample space.
Think about how are these probabilities related?
Use the coin and spinner f rom the "Coin and Spinner" worksheet that y ou completed earlier.
36
Think
about it.
Coin
6
Apply
your
knowledg
e
7
P(Tails) =
Spinner
1
2
P(Red) =
Coin and Spinner
1
4
1
8
P( Tails and Red ) =
Check this answer by using y our sample space.
These should be related in the same way.
Based on your results of question 5, if you know the probability of 2 independent events,
how can you find the probability of those 2 events happening together?
Multiply the two probabilities together.
8
9
10
11
12
If P(A) =
3
4
and P(B) =
5
7
then what is the probability P(A and B) ?
What about 3 independent events, such as flipping 3 coins?
Would you make a table display the sample space or would you make a tree diagram?
Use a Separate sheet of paper to construct an appropriate sample space for tossing 3 coins.
Then use your sample space to answer the following questions.
P(T,T,T) =
1
8
14
Use the Fundamental Counting Principle to compute this one.
P(3 of the same in a row) =
2
8
P(2 heads and a tail in that order) =
1
8
P(2 heads and a tail in any order) =
3
8
Extend
13
15
28
Y ou may need to use your tree diagram f or
questions 10-12.
Click the LINK and scroll down to see a
three-coin tree diagram
Another way to write the sample space is
to list all of the possible outcomes.
Click the f ollowing link and scroll down to example 2
to f ind what y ou are looking f or.
LINK
Here
Imagine taking a 5-question True / False Test for which you have not studied.
14a. Find the probability that you will get the first question correct.
1
2
14b. Find the probability that you will get all 5 questions correct.
1
32
14c. Find the probability that you will pass, if you need 3 out of 5 to pass.
10
32
The probability of Acing
the test by guessing
is v ery , v ery low.
This is a v ery dif f icult
question. There's more
than 1 way to get 3
right.
Sample Space for 2 independent events, a coin and a
spinner
#1.
C
O
I
N
4 color spinner
4
H
R,H
B,H
G,H
Y,H
T
R,T
B,T
G,T
Y,T
Red
1
#2
3
Complete the Sample space in table form.
2
Blue
Green
Spinner
Complete the Sample space in tree diagram form.
Try this link if you need to see another example of a tree diagram
SPINNER
COIN
H
1
T
3. What is the total number of outcomes for the
flipping of the coin?
2
4. What is the total number of outcomes for the
spinning of the spinner?
H
2
T
4
5. What is the total number of outcomes for
both events together?
H
8
3
T
6. For the coin only find P(H).
P(H) =
1
2
7. For the spinner find P(BLUE).
H
4
P(BLUE) =
1
4
8. For both together find P(Blue and Head)
P(Blue and Head) =
1
8
Yellow
T
Link
Fill in the sample space for rolling 2 dice.
Use the sum of the 2 dice.
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2
3
4
5
6
7
3
4
5
6
7
8
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Questions
1. Use the sample space to find P(6).
P(6) =
2. Use the sample space to find P(10).
P(10) =
3. Use the sample space to find P(double).
P(double) =
4. Which sum has the highest probability?
4
5
6
7
8
9
5
6
7
8
9
10
6
7
8
9
10
11
7
8
9
10
11
12
find
In the chart below, use the Fill tool to draw a colorful bar
graph of the sample space of rolling 2 dice.
8
7
6
5
4
3
2
1
2
3
4
5
6
7
8
9
Sum of 2 dice
5
36
3
36
6
36
7
5. I could win a game by rolling either a sum of 7 or 11,
the probability that I will win.
How many of each sum
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☻
☻
☻
10
11
12
P(7 or 11) =
8
36