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Experimental Probability

? ESSENTIAL QUESTION How can you use experimental probability to solve real-world problems?

12 MODULE

LESSON 12.1

Probability

7.SP.5, 7.SP.7a

LESSON 12.2

Experimental Probability of Simple Events

7.SP.6, 7.SP.7b

LESSON 12.3

Experimental Probability of Compound Events

7.SP.8, 7.SP.8a, 7.SP.8b, 7.SP.8c

LESSON 12.4

Making Predictions with Experimental Probability

7.SP.6

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Real-World Video

Meteorologists use sophisticated equipment to gather data about the weather. Then they use experimental probability to forecast, or predict, what the weather conditions will be.

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Animated Math

Interactively explore key concepts to see how math works.

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363

Are YOU Ready?

Complete these exercises to review skills you will need for this module.

Simplify Fractions

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EXAMPLE

Simplify _12_21.

12: 1, 2, 3, 4, 6, 12 List all the factors of the numerator and denominator. 21: 1, 3, 7, 21 Circle the greatest common factor (GCF).

_1_2_?_3_ 21 ? 3

=

_ 4 7

Divide the numerator and denominator by the GCF.

Write each fraction in simplest form.

1.

_6_ 10

2.

_9_ 15

3.

_1_6 24

4.

_9_ 36

5.

_4_5 54

6.

_3_0 42

7.

_3_6 60

8.

_1_4 42

Write Fractions as Decimals

EXAMPLE

_1_3 25

25 130..0502

- 12.5

50

- 50

0

Write the fraction as a division problem. Write a decimal point and a zero in the dividend. Place a decimal point in the quotient. Write more zeros in the dividend if necessary.

Write each fraction as a decimal.

9.

_ 3 4

10.

_ 7 8

Percents and Decimals

11.

_3_ 20

12.

_1_9 50

EXAMPLE

109% = 100% + 9%

=

_1_00_ 100

+

__9 _ 100

= 1 + 0.09

= 1.09

Write each percent as a decimal.

13. 67%

14. 31%

Write each decimal as a percent.

17. 0.13

18. 0.55

Write the percent as the sum of 1 whole and a percent remainder. Write the percents as fractions. Write the fractions as decimals. Simplify.

15. 7% 19. 0.08

16. 146% 20. 1.16

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Reading Start-Up

Visualize Vocabulary

Use the words to complete the graphic. You can put more than one word in each box.

3:4, 75%

facts used to make decisions act of collecting facts

Making Mathematical

Predictions

Understand Vocabulary

Match the term on the left to the definition on the right.

1. probability

A. Measures the likelihood that the event will occur.

2. trial

B. A set of one or more outcomes.

3. event

C. Each observation of an experiment.

Vocabulary

Review Words

data (datos) observation (observaci?n) percent (porcentaje) ratio (raz?n)

Preview Words

complement (complemento) compound event (suceso compuesto) event (suceso) experiment (experimento) experimental probability (probabilidad experimental) outcome (resultado) probability (probabilidad) simple event (suceso simple) simulation (simulaci?n) trial (prueba)

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Active Reading

Pyramid Before beginning the module, create a rectangular pyramid to help you organize what you learn. Label each side with one of the lesson titles from this module. As you study each lesson, write important ideas, such as vocabulary, properties, and formulas, on the appropriate side.

Module 12 365

GETTING READY FOR

Experimental Probability

Understanding the standards and the vocabulary terms in the standards will help you know exactly what you are expected to learn in this module.

7.SP.6

Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability.

Key Vocabulary

simple event (suceso simple) An event consisting of only one outcome.

experimental probability (probabilidad experimental) The ratio of the number of times an event occurs to the total number of trials, or times that the activity is performed.

What It Means to You

You will use experimental probabilities to make predictions and solve problems.

EXAMPLE 7.SP.6

Caitlyn finds that the experimental probability of her making a goal in hockey is 30%. Out of 500 attempts to make a goal, about how many could she predict she would make?

_3_ 10

?

500

=

x

150 = x

Caitlyn can predict that she will make about 150 of the 500 goals that she attempts.

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7.SP.7b

Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.

Key Vocabulary

sample space (espacio muestral) All possible outcomes of an experiment.

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366 Unit 6

What It Means to You

You will use data to determine experimental probabilities.

EXAMPLE 7.SP.7b

Anders buys a novelty coin that is weighted more heavily on one side. He flips the coin 60 times and a head comes up 36 times. Based on his results, what is the experimental probability of flipping a head?

experimental

probability

=

n_u_m__b_e_r _o_f _ti_m_e_s_e_v_e_n_t _o_cc_u_r_s total number of trials

=

_3_6 60

=

_ 3 5

The experimental probability of flipping a head is _35.

LESSON

12.1 Probability

7.SP.5

Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event

occurring. Larger numbers indicate greater

likelihood. A probability near 0 indicates iannduicnaltikeeslayneveevnetn,tatphraotbiasbnieliittyhaerrouunnldik_12ely

nor likely, and a probability near 1 indicates

a likely event. Also 7.SP.7a

? ESSENTIAL QUESTION How can you describe the likelihood of an event?

EXPLORE ACTIVITY

7.SP.5

Finding the Likelihood of an Event

Each time you roll a number cube, a number from 1 to 6 lands face up. This is called an event.

Work with a partner to decide how many of the six possible results of rolling a number cube match the described event.

Then order the events from least likely (1) to most likely (9) by writing a number in each box to the right.

Rolling a number less than 7

Rolling an 8

Rolling a number greater than 4

Rolling a 5

Rolling a number other than 6

Rolling an even number

Rolling a number less than 5

Rolling an odd number

Rolling a number divisible by 3

Reflect

1. Are any of the events impossible?

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Lesson 12.1 367

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Describing Events

An experiment is an activity involving chance in which results are observed. Each observation of an experiment is a trial, and each result is an outcome. A set of one or more outcomes is an event.

The probability of an event, written P(event), measures the likelihood that the event will occur. Probability is a measure between 0 and 1 as shown on the number line, and can be written as a fraction, a decimal, or a percent.

If the event is not likely to occur, the probability of the event is close to 0. If an event is likely to occur, the event's probability is closer to 1.

Impossible

Unlikely

As likely as not

Likely Certain

0

_ 1 2

1

0

0.5

1.0

0%

50%

100%

EXAMPLE 1

7.SP.5

Tell whether each event is impossible, unlikely, as likely as not, likely, or

certain.

Then,

tell

whether

the

probability

is

0,

close

to

0,

_ 1

2

,

close

to

1,

or

1.

A You roll a six-sided number cube and the number is 1 or greater.

This event is certain to happen. Its probability is 1.

Because you can roll the numbers 1, 2, 3, 4, 5, and 6 on a number cube, there are 6 possible outcomes.

B You roll two number cubes and the sum of the numbers is 3.

This event is unlikely to happen. Its probability is close to 0.

C A bowl contains disks marked with the numbers 1 through 10. You close your eyes and select a disk at random. You pick an odd number.

This

event

is

as

likely

as

not.

The

probability

is

_ 1

2

.

Math Talk

D A spinner has 8 equal sections marked 0 through 7. You spin and land on a

Mathematical Practices

prime number.

Is an event that is not

certain an impossible event? Explain.

This

event

is

as

likely

as

not.

The

probability

is

_ 1

2

.

Remember that a prime number is a whole number greater than 1 and has exactly 2 divisors, 1 and itself.

Reflect

2.

The

probability

of

event

A

is

_ 1

3

.

The

probability

of

event

B

is

_ 1

4

.

What

can

you conclude about the two events?

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368 Unit 6

YOUR TURN

3. A hat contains pieces of paper marked with the numbers 1 through 16.

Tell whether picking an even number is impossible, unlikely, as likely as

not,

likely,

or

certain.

Tell

whether

the

probability

is

0,

close

to

0,

_ 1

2

,

close

to 1, or 1.

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Finding Probability

The sample space is a set of all possible outcomes for an event. A sample space can be small, such as the 2 outcomes when a coin is flipped. Or a sample space can be large, such as the possible number of Texas Classic automobile license plates. Identifying the sample space can help you calculate the probability of an event.

Probability of An Event

P(event)

=

_______n_u_m__b_e_r_o_f_t_im__e_s_t_h_e_e_v_e_n__t _o_c_c_u_rs_______ total number of equally likely possible outcomes

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EXAMPLE 2

What is the probability of rolling an even number on a standard number cube?

STEP 1 Find the sample space for a standard number cube.

{1, 2, 3, 4, 5, 6}

There are 6 possible outcomes.

STEP 2 Find the number of ways to roll an even number.

2, 4, 6

The event can occur 3 ways.

STEP 3 Find the probability of rolling an even number.

P(even)

=

_nu_m__b_e_r_o_f _w_a_y_s _to__ro_l_l a_n__ev_e_n__n_u_m_b_e_r number of faces on a number cube

=

_ 3 6

=

_ 1 2

Substitute values and simplify.

The

probability

of

rolling

an

even

number

is

_ 1

2

.

7.SP.7a

Lesson 12.1 369

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YOUR TURN

Find each probability. Write your answer in simplest form.

4. Picking a purple marble from a jar with 10 green and 10 purple

5. Rolling a number greater than 4 on a standard number cube.

marbles.

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Using the Complement of an Event

The complement of an event is the set of all outcomes in the sample space that are not included in the event. For example, in the event of rolling a 3 on a number cube, the complement is rolling any number other than 3, which means the complement is rolling a 1, 2, 4, 5, or 6.

An Event and Its Complement

The sum of the probabilities of an event and its complement equals 1. P(event) + P(complement) = 1

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370 Unit 6

You can apply probabilities to situations involving random selection, such as drawing a card out of a shuffled deck or pulling a marble out of a closed bag.

EXAMPLE 3

7.SP.7a

There are 2 red jacks in a standard deck of 52 cards. What is the probability of not getting a red jack if you select one card at random?

P(event) + P(complement) = 1

P(red jack) + P(not a red jack) = 1

_2_ 52

+

P(not

a

red

jack)

=

1

_2_ 52

+

P(not

a

red

jack)

=

_5_2 52

-_52_2

-_52_2

P(not

a

red

jack)

=

_5_0 52

P(not

a

red

jack)

=

_2_5 26

The probability of getting a red jack is _52_2_.

Substitute

_2__

52

for

P(red

jack).

Subtract

_2__

52

from

both

sides.

Simplify.

The probability that you will not draw a red jack is _22_65. It is likely that you will not select a red jack.

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