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Learn to find the probabilities of independent

and dependent events.

Vocabulary

independent events 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.

Additional Example 1 Classifying Events as

Independent or Dependent

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

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.

Try This Example 1

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

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.

(No Transcript)

Additional Example 2A Finding the Probability of

Independent 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?

The outcome of each choice does not affect the

outcome of the other choices, so the choices are

independent.

Multiply.

P(blue, blue, blue)

0.125

Additional Example 2B Finding the Probability of

Independent Events

B. What is the probability of choosing a blue

marble, then a green marble, and then a blue

marble?

Multiply.

P(blue, green, blue)

0.125

Additional Example 2C Finding the Probability of

Independent Events

C. What is the probability of choosing at least

one blue marble?

Think P(at least one blue) P(not blue, not

blue, not blue) 1.

P(not blue, not blue, not blue)

0.125

Multiply.

Subtract from 1 to find the probability of

choosing at least one blue marble.

1 0.125 0.875

Try This Example 2A

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?

The outcome of each choice does not affect the

outcome of the other choices, so the choices are

independent.

Multiply.

P(blue, blue)

0.0625

Try This Example 2B

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?

Multiply.

P(blue, red)

0.0625

Try This Example 2C

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?

Think P(at least one blue) P(not blue, not

blue) 1.

P(not blue, not blue)

0.5625

Multiply.

Subtract from 1 to find the probability of

choosing at least one blue marble.

1 0.5625 0.4375

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.

Additional Example 3A Find the Probability of

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?

P(first consonant)

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)

Multiply.

Additional Example 3B Find the Probability of

Dependent Events

B. If two letters are chosen at random, what is

the probability that they will both be consonants

or both be vowels?

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.

P(first vowel)

If the first letter chosen was a vowel, there are

now only 2 vowels and 8 total letters left in the

box.

Additional Example 3B Continued

Find the probability that the second letter

chosen is a vowel.

P(second vowel)

Multiply.

The events of both consonants and both vowels are

mutually exclusive, so you can add their

probabilities.

P(consonant) P(vowel)

Try This Example 3A

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?

P(first consonant)

Try This Example 3A Continued

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.

P(second consonant)

Multiply.

Try This Example 3B

B. If two letters are chosen at random, what is

the probability that they will both be consonants

or both be vowels?

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.

P(first vowel)

If the first letter chosen was a vowel, there are

now only 3 vowels and 8 total letters left in the

box.

Try This Example 3B Continued

Find the probability that the second letter

chosen is a vowel.

P(second vowel)

Multiply.

The events of both consonants and both vowels are

mutually exclusive, so you can add their

probabilities.

P(consonant) P(vowel)

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?

dependent

independent

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