Learn to find the probabilities of independent and dependent events.

Course 3

10-5 Independent and Dependent Events

Vocabulary

compound events

independent eventsdependent events

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Course 3

10-5 Independent and Dependent Events

A compound event is made up of one or more separate events. To find the probability of a compound event, you need to know if the events are independent or dependent.

Course 3

10-5 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.

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

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.

Course 3

10-5 Independent and Dependent Events

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

Course 3

10-5 Independent and Dependent Events

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 2

P(not blue, not blue, not blue) =12

· 12

· 12

= 18

= 0.125 Multiply.

Course 3

10-5 Independent and Dependent Events

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

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

Check It Out: Example 2B

In each box, P(blue) = . 14

P(blue, red) = 14

· 14

= 116

= 0.0625 Multiply.

In each box, P(red) = . 14

Course 3

10-5 Independent and Dependent Events

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

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

Check It Out: 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.Course 3

10-5 Independent and Dependent Events

Course 3

10-5 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.

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

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) =

Course 3

10-5 Independent and Dependent Events

23

69

=

Because the first letter is not replaced, the sample space is different for the second letter, so the events are dependent. Find the probability that the first letter chosen is a consonant.

Additional Example 3A Continued

Course 3

10-5 Independent and Dependent Events

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.

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.

Course 3

10-5 Independent and Dependent Events

P(second vowel) = 14

28

=

1 1214

13

· = Multiply.

12

5 12

1 12

+ = 6 12

=

The probability of getting two letters that are either both consonants or both vowels is . 1

2

P(consonant) + P(vowel)

Check It Out: 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.

Course 3

10-5 Independent and Dependent Events

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 . 4

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