Probability is the likelihood that a particular event will occur. We express the
probability of an event occurring using the numbers from 0 through 1. The numbers between 0 and 1 can be written as fractions, decimals, or percents. Thinking Skill
Connect
How do you write 0, 12, and 1 as a decimal and percent?
• A probability of 0 represents an event that cannot occur or is
impossible.
• A probability of 1 represents an event that is certain to occur.
• A probability of 12 represents an event that is equally likely to occur as to not occur.
• A probability less than 12 means the event is unlikely to occur. • A probability greater than 12 means the event is likely to occur.
Unlikely
Impossible Equally Likely Certain
Likely
1
2 1
0
We can use this formula to find the probability of an event occurring.
Probability (Event) = number of favorable outcomes total number of possible outcomes
Suppose we have a bag that contains 4 red marbles and 5 blue marbles. We want to find the probability of picking one marble of a specific color from the bag without looking.
Reading Math The symbols
P(Red) and P(R)
are both read as “the probability of red.”
Generalize How can we use the formula to find the probability of picking each color?
P(Red) =
number of red marbles total number of marbles P(R) = 4
9
We find that the probability of picking red is 49.
P(Blue) =
number of blue marbles total number of marbles P(B) = 5
9
We find that the probability of picking blue is 59.
We can also write the probability of picking a green marble from this bag of marbles.
P(Green) =
number of green marbles total number of marbles =
0 9= 0 The probability of picking green is 0.
Example 3
This number cube has 1 through 6 dots on the faces of the cube. If the number cube is rolled once, what is the probability of each of these outcomes?
a. rolling a 4
b. rolling a number greater than 4 c. rolling a number greater than 6 d. rolling a number less than 7
Solution
Since there are six different faces on the number cube, there are six equally likely outcomes. Thus, there are six possible outcomes.
a. There is only one way to roll a 4 with the number cube. The probability of rolling a 4 is 16.
b. The numbers greater than 4 on the number cube are 5 and 6, so there are two ways to roll a number greater than 4. The probability of rolling a number greater than 4 is 26.
c. There are no numbers greater than 6 on the number cube. So it is impossible to roll a number greater than 6. The probability of rolling a number greater than 6 is 0
6 or 0.
d. There are six numbers less than 7 on the number cube. So there are six ways to roll a number less than 7. The probability of rolling a number less than 7 is 66 or 1.
Example 4
This spinner is divided into five equal sectors and is numbered 1 through 5. The arrow is spun once.
a. How many different outcomes are possible?
b. What is the probability of spinning a 3? c. What is the probability of not spinning a 3?
Solution
The probability that the spinner will stop in a given sector is equal to the fraction of the spinner’s face occupied by that sector.
a. There are five equally likely outcomes when spinning this spinner.
1
3 2 4
Lesson 14 97 b. Spinning a 3 is one of five equally likely outcomes. We can use the
formula to find the probability of spinning a 3.
P(3) =
number of favorable outcomes total number of possibles outcomes P(3) = 1
5
The probability of spinning a 3 is 15.
c. We can also use the formula to find the probability of not spinning a 3. There are four ways for the spinner not to stop on 3.
P(not 3) =
number of favorable outcomes total number of possibles outcomes P(not 3) = 4
5
The probability of not spinning a 3 is 45.
Notice that the sum of the probability of an event occurring plus the probability of the event not occurring is 1.
P(3) + P(not 3) = 1 1 5+ 4 5 = 5 5 or 1
When the sum of the probabilities of two events is equal to 1, they are called
complementary events.
Example 5
The spinner at the right is divided into one half and two fourths. What is the probability of the spinner stopping on 3?
Solution
There are three possible outcomes, but the outcomes are not equally likely, because the sizes of the regions are not all equal.
• Since Region 1 is one half of the whole area, the probability of the spinner stopping on 1 is 12.
• Regions 2 and 3 each represent 14 of the whole area. The probability of the spinner stopping on 2 is 14, and the probability of it stopping on 3 is also 14.
Generalize The probability of the spinner stopping on 2 or 3 is 12. What is the complement of this event?
1 2
Practice Set
Along with each answer, include the equation you used to solve the problem. a. Only 39% of the lights were on. What percent of the lights wereoff?
b. Two fifths of the students did not go to the museum. What fraction of the students did go to the museum?
c. Write a word problem about parts of a whole that fits this equation: 45% + g = 100%
d. Rolling a number cube once, what is the probability of rolling a number less than 4?
Analyze This spinner is divided into four equal sections.
e. What is the probability of this spinner stopping on 3?
f. What is the probability of this spinner stopping on 5?
g. What is the probability of this spinner stopping on a number less than 6?
This spinner is divided into one half and two fourths.
h. What is the probability of this spinner stopping on A?
i. What is the probability of this spinner not stopping on B?
Written Practice
Strengthening Concepts * 1.(11)
The USDA recommends that adults eat at least 85 grams of whole grain products each day. Ryan ate 63 grams of whole-grain cereal. How many more grams of whole grain products should he eat?
* 2.
(14)
Seven tenths of the new recruits did not like their first haircut. What fraction of the new recruits did like their first haircut?
* 3.
(12)
The Declaration of Independence was signed in 1776. The U.S. Constitution was ratified in 1789. How many years passed between these two events?
* 4.
(13) Formulate
Write a word problem that fits this equation: 12p = $2.40
* 5. In 2000, nearly 18% of cars sold in North America were silver. What
1 2 3 4 A B C
Lesson 14 99 * 6.
(10)
Model Draw and shade circles to show that 31
3⫽103.
7.
(5) Use digits to write four hundred seven million, forty-two thousand, six
hundred three. 8.
(2) Analyze What property is illustrated by this equation?
3 ∙ 2 ∙ 1 ∙ 0 = 0 9.
(6) a. List the common factors of 40 and 72.
b. What is the greatest common factor of 40 and 72? 10.
(7) Name three segments in the figure below in order of length from
shortest to longest.
W X Y
11.
(8) Describe how to find the fraction of the group
that is shaded. Solve: 12. (3) b − 407 = 623 13. (3) $20 − e = $3.47 14. (3) 7 ∙ 5f = 7070 15. (3) m 25= 25 16.(3) 5 8 7 6 5 9 4 3 6 4 7 8 5 n ⫹ 6 89 17. (3) a + 295 = 1000 Simplify: * 18. (10) 335⫹ 245 * 19. (10) 5 2∙ 32 20. (1) $3.63 + $0.87 + 96¢ 21. (9) 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1 * 22. (9) 2 3∙ 23∙ 23 23.(1) 900 20 24. (1) 145 ⫻ 74 25. (1) 30(65¢) 26. (2) (5)(5 + 5) 27. (4) 9714 − 13,456 28. (7)
Classify Name each type of angle illustrated:
a. b. c. * 29. (9) How many 4 5 s are in 1? 30.
(14) Rolling a number cube once, what is the probability of rolling a number