Download Problem 1 What Are the Chances?

Problem 1 What Are the Chances?
A six-sided number cube has one number, from 1 through 6, on each of its faces. Number
cubes are often used when playing many board games.
1. Create a list of all the possible numbers that can appear on the top face if you roll a
six-sided number cube.
The numbers 1, 2, 3, 4, 5, and 6 are the outcomes that can occur when rolling a six-sided
number cube. An outcome is the result of a single trial of an experiment. An experiment
is a situation involving chance that leads to results, or outcomes. A list of all possible
outcomes of an experiment is called a sample space. A sample space is typically
enclosed in brackets, { }, with commas between the outcomes.
An event is one or a group of possible outcomes for a given situation. A simple event is
an event consisting of one outcome. For example, in the number cube experiment, an
event could be rolling an even number. However, rolling a 5 is a simple event.
Probability is a measure of the likelihood that an event will occur. It is a way of assigning a
numerical value to the chance that an event will occur. The probability of an event is often
written as P(event). For example, in the number cube experiment, the probability of rolling
a 5 could be written as P(rolling a 5), or P(5). The probability of rolling an even number
could be written as P(rolling an even number), or P(even).
In general, the way to determine the probability of an event is:
an event
can occur
    
   
 ​.
probability 5 _________________________________
​ number of times
number of possible outcomes
830 • Chapter 16 Introduction to Probability
© 2011 Carnegie Learning
2. List the sample space for the experiment of rolling a six-sided number cube.
To determine the probability of rolling an odd number, or P(odd), follow these steps:
Step 1: First, list all the possible outcomes.
The possible odd numbers that can be rolled are 1, 3, and 5.
Step 2: Add the number of outcomes.
There are 3 possible outcomes of rolling an odd number.
Step 3: Use the equation to determine the probability of rolling an odd number.
3 times of rolling an odd number
an
even can occur
    
   
    ​
 ​5 ______________________________
​ 
P(odd) 5 ________________________________
​ number of times
number of possible outcomes
6 number of possible outcomes
The probability of rolling an odd number is 3 out of 6 outcomes.
3. What is the probability of rolling a 4, or P(4)? Explain your reasoning.
So, to determine
P(4) consider how many
times a 4 can occur
when you roll a six-sided
number cube.
4. What is the probability of rolling a 6, or P(6)? Explain
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your reasoning.
16.1 Defining and Representing Probability • 831
5. Determine the probability of rolling an even number.
a. Which outcome or outcomes make up the event of rolling an even number?
b. Calculate the probability of rolling an even number.
6. Determine the probability of rolling a number greater than 4.
a. Which outcome(s) make up the event of rolling a number greater than 4?
b. Calculate the probability of rolling a number greater than 4.
© 2011 Carnegie Learning
832 • Chapter 16 Introduction to Probability
Problem 3 Even I Win, Odd You Win...
Joel and Hiram are playing a game using 2 six-sided number cubes. The number cubes
are rolled, and the sum of the 2 numbers shown is calculated. If the sum is even, Joel
wins a point. If the sum is odd, Hiram wins a point.
1. List the sample space for the game.
2. Do you think each of the outcomes is equally likely?
Explain your reasoning.
Time to
make your
predictions.
3. Who do you think has a better chance of winning this game?
© 2011 Carnegie Learning
Explain your reasoning.
16.2 Determining Experimental Probability • 847
4. Play the game 25 times with a partner. Record your sums in the table.
Sum of Two
Number Cubes
Tally Mark
Sum of Two
Number Cubes
2
8
3
9
4
10
5
11
6
12
Tally Mark
7
5. According to your experiment, who would win the game between Joel and Hiram?
6. Calculate the experimental probability for each outcome. Express your probability as
a. P(2) 5
b. P(3) 5
c. P(4) 5
d. P(5) 5
e. P(6) 5
g. P(8) 5
i. P(10) 5
f. P(7) 5
h. P(9) 5
k. P(12) 5
848 • Chapter 16 Introduction to Probability
j. P(11) 5
© 2011 Carnegie Learning
a fraction and a percent.
Problem 1 Double Your Fun—Again!
Previously, Joel and Hiram were playing a game using 2 six-sided number cubes. The
number cubes were rolled, and the sum of the 2 numbers shown was calculated. If the
sum was even, Joel won a point. If the sum was odd, Hiram won a point. The experimental
probability (the ratio of the number of times an event occurs to the total number of trials
performed) was then calculated.
When you calculate the actual probability of an event before performing the experiment,
you are determining the theoretical probability. Theoretical probability is the
mathematical calculation that an event will happen in theory. The formula for theoretical
probability is the ratio of the number of desired outcomes to the total number of possible
outcomes, or the sample space.
number of
desired
outcomes  ​
    
   
Theoretical Probability 5 ​ ________________________________
total number of possible outcomes
1. List the sample space for the game played by Joel and Hiram.
In the previous experiment of rolling 2 six-sided number cubes, each of the outcomes in
the sample space is not equally likely. One method to determine the probabilities of the
outcomes is to make a list of all the possibilities.
For example, (1, 1) could represent rolling a 1 on the first number cube and a 1 on the
second number cube. Another example, (1, 2) could represent rolling a 1 on the first
number cube and a 2 on the second number cube.
The list shown lists all the possibilities when rolling 2 six-sided number cubes.
(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6),
(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6),
(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6),
(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6),
(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)
2. How many different possibilities are there when rolling 2 six-sided number cubes?
852 • Chapter 16 Introduction to Probability
© 2011 Carnegie Learning
(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6),
Making a list can sometimes take a lot of time. Another way to determine all of the
possibilities is to use a number array.
3. The array shown has two numbers filled in: 2 and 7. What does the 2 mean in the
array? What does the 7 mean in the array?
Number Cube 1
1
1
2
3
4
5
6
2
Number Cube 2
2
3
4
7
5
6
4. Complete the number array shown.
© 2011 Carnegie Learning
5. How many different possibilities are in the number array?
6. Does it appear that the list of all the possibilities when rolling 2 six-sided number
cubes has the same number of possibilities as the number array?
16.3 Determining Theoretical Probability • 853
7. Using the number array, determine the number of times each sum appears.
a. sum of 2
b. sum of 3
c. sum of 4
d. sum of 5
e. sum of 6
f. sum of 7
g. sum of 8
h. sum of 9
i. sum of 10
j. sum of 11
k. sum of 12
8. Calculate the theoretical probabilities for each sum.
a. P(2) 5
b. P(3) 5
c. P(4) 5
d. P(5) 5
e. P(6) 5
f. P(7) 5
g. P(8) 5
h. P(9) 5
i. P(10) 5
j. P(11) 5
k. P(12) 5
9. Is there an equally likely chance for each number to result from rolling 2 six-sided
10. According to the theoretical probabilities, who should win the game?
Explain your reasoning.
854 • Chapter 16 Introduction to Probability
© 2011 Carnegie Learning
number cubes according to the theoretical probabilities?