Pre-Algebra

9-7 Independent and Dependent Events9-7 Independent and Dependent

Events

Pre-Algebra

Warm UpWarm Up

Problem of the DayProblem of the Day

Lesson PresentationLesson Presentation

Pre-Algebra

9-7 Independent and Dependent Events

Targeted Instruction:

Write the following equation in slope-intercept form (y=mx+b) and graph using the slope and intercept.

7x – 6y = 18

Pre-Algebra

9-7 Independent and Dependent Events

Warm UpEvaluate each expression.

1. 8!

2.

3. Find the number of permutations of the letters in the word quiet if no letters are used more than once.

40,320

720

120

Pre-Algebra

9-7 Independent and Dependent Events

10!7!

Pre-Algebra

9-7 Independent and Dependent Events

Learn to find the probabilities of independent and dependent events.

Pre-Algebra

9-7 Independent and Dependent Events

Vocabulary

independent eventsdependent events

Pre-Algebra

9-7 Independent and Dependent Events

Events are independent events if the occurrence of one event does not affect the probability of the other. Events are dependent events if the occurrence of one does affect the probability of the other.

Pre-Algebra

9-7 Independent and Dependent Events

Determine if the events are dependent or independent.

A. getting tails on a coin toss and rolling a 6 on a number cube

B. getting 2 red gumballs out of a gumball machine

Additional Example 1: Classifying Events as Independent or Dependent

Tossing a coin does not affect rolling a number cube, so the two events are independent.

After getting one red gumball out of a gumball machine, the chances for getting the second red gumball have changed, so the two events are dependent.

Pre-Algebra

9-7 Independent and Dependent Events

Determine if the events are dependent or independent.

A. rolling a 6 two times in a row with the same number cube

B. a computer randomly generating two of the same numbers in a row

Try This: Example 1

The first roll of the number cube does not affect the second roll, so the events are independent.

The first randomly generated number does not affect the second randomly generated number, so the two events are independent.

Pre-Algebra

9-7 Independent and Dependent Events

Pre-Algebra

9-7 Independent and Dependent Events

Three separate boxes each have one blue marble and one green marble. One marble is chosen from each box.

A. What is the probability of choosing a blue marble from each box?

Additional Example 2A: Finding the Probability of Independent Events

The outcome of each choice does not affect the outcome of the other choices, so the choices are independent.

P(blue, blue, blue) =

In each box, P(blue) = .12

12

· 12

· 12

= 18

= 0.125 Multiply.

Pre-Algebra

9-7 Independent and Dependent Events

B. What is the probability of choosing a blue marble, then a green marble, and then a blue marble?

Additional Example 2B: Finding the Probability of Independent Events

P(blue, green, blue) = 12

· 12

· 12

= 18

= 0.125 Multiply.

In each box, P(blue) = .12

In each box, P(green) = .1 2

Pre-Algebra

9-7 Independent and Dependent Events

C. What is the probability of choosing at least one blue marble?

Additional Example 2C: Finding the Probability of Independent Events

1 – 0.125 = 0.875

Subtract from 1 to find the probability of choosing at least one blue marble.

Think: P(at least one blue) + P(not blue, not blue, not blue) = 1.

In each box, P(not blue) = .1 2P(not blue, not blue, not blue) =

12

· 12

· 12

= 18

= 0.125 Multiply.

Pre-Algebra

9-7 Independent and Dependent Events

Two boxes each contain 4 marbles: red, blue, green, and black. One marble is chosen from each box.

A. What is the probability of choosing a blue marble from each box?

Try This: Example 2A

The outcome of each choice does not affect the outcome of the other choices, so the choices are independent.

In each box, P(blue) = .14

P(blue, blue) = 14

· 14

= 116

= 0.0625 Multiply.

Pre-Algebra

9-7 Independent and Dependent Events

Two boxes each contain 4 marbles: red, blue, green, and black. One marble is chosen from each box.

B. What is the probability of choosing a blue marble and then a red marble?

Try This: Example 2B

In each box, P(blue) = .14

P(blue, red) = 14

· 14

= 116

= 0.0625 Multiply.

In each box, P(red) = .14

Pre-Algebra

9-7 Independent and Dependent Events

Two boxes each contain 4 marbles: red, blue, green, and black. One marble is chosen from each box.

C. What is the probability of choosing at least one blue marble?

Try This: Example 2C

In each box, P(blue) = .14

P(not blue, not blue) = 34

· 34

= 916

= 0.5625 Multiply.

Think: P(at least one blue) + P(not blue, not blue) = 1.

1 – 0.5625 = 0.4375Subtract from 1 to find the probability of choosing at least one blue marble.

Pre-Algebra

9-7 Independent and Dependent Events

To calculate the probability of two dependent events occurring, do the following:

1. Calculate the probability of the first event.

2. Calculate the probability that the second event would occur if the first event had already occurred.

3. Multiply the probabilities.

Pre-Algebra

9-7 Independent and Dependent Events

The letters in the word dependent are placed in a box.

A. If two letters are chosen at random, what is the probability that they will both be consonants?

Additional Example 3A: Find the Probability of Dependent Events

P(first consonant) = 23

69

=

Pre-Algebra

9-7 Independent and Dependent Events

Additional Example 3A Continued

If the first letter chosen was a consonant, now there would be 5 consonants and a total of 8 letters left in the box. Find the probability that the second letter chosen is a consonant.

P(second consonant) = 58

5 12

58

23

· =

The probability of choosing two letters that are both consonants is .5

12

Multiply.

Pre-Algebra

9-7 Independent and Dependent Events

B. If two letters are chosen at random, what is the probability that they will both be consonants or both be vowels?

Additional Example 3B: Find the Probability of Dependent Events

There are two possibilities: 2 consonants or 2 vowels. The probability of 2 consonants was calculated in Example 3A. Now find the probability of getting 2 vowels.

Find the probability that the first letter chosen is a vowel.

If the first letter chosen was a vowel, there are now only 2 vowels and 8 total letters left in the box.

P(first vowel) = 13

39

=

Pre-Algebra

9-7 Independent and Dependent Events

Additional Example 3B Continued

Find the probability that the second letter chosen is a vowel.

The events of both consonants and both vowels are mutually exclusive, so you can add their probabilities.

P(second vowel) = 14

28

=

1 12

14

13

· = Multiply.

12

5 12

1 12

+ = 6 12

=

The probability of getting two letters that are

either both consonants or both vowels is .12

P(consonant) + P(vowel)

Pre-Algebra

9-7 Independent and Dependent Events

The letters in the phrase I Love Math are placed in a box.

A. If two letters are chosen at random, what is the probability that they will both be consonants?

Try This: Example 3A

P(first consonant) = 59

Pre-Algebra

9-7 Independent and Dependent Events

Try This: Example 3A Continued

P(second consonant) =

5 18

12

59

· =

The probability of choosing two letters that are both consonants is .5

18

Multiply.

If the first letter chosen was a consonant, now there would be 4 consonants and a total of 8 letters left in the box. Find the probability that the second letter chosen is a consonant.

12

48

=

Pre-Algebra

9-7 Independent and Dependent Events

B. If two letters are chosen at random, what is the probability that they will both be consonants or both be vowels?

Try This: Example 3B

There are two possibilities: 2 consonants or 2 vowels. The probability of 2 consonants was calculated in Try This 3A. Now find the probability of getting 2 vowels.

Find the probability that the first letter chosen is a vowel.

If the first letter chosen was a vowel, there are now only 3 vowels and 8 total letters left in the box.

P(first vowel) = 49

Pre-Algebra

9-7 Independent and Dependent Events

Try This: Example 3B Continued

Find the probability that the second letter chosen is a vowel.

The events of both consonants and both vowels are mutually exclusive, so you can add their probabilities.

P(second vowel) = 38

12 72

38

49

· = Multiply.16

=

49

5 18

1 6

+ = 8 18

= P(consonant) + P(vowel)

The probability of getting two letters that are

either both consonants or both vowels is .49

Pre-Algebra

9-7 Independent and Dependent Events

Lesson Quiz

Determine if each event is dependent or independent.

1. drawing a red ball from a bucket and then drawing a green ball without replacing the first

2. spinning a 7 on a spinner three times in a row

3. A bucket contains 5 yellow and 7 red balls. If 2 balls are selected randomly without replacement, what is the probability that they will both be yellow?

independent

dependent

533

Pre-Algebra

9-7 Independent and Dependent Events

Homework/Practice

Textbook page 480 (7-14)