**3 4** 2 5 3 5 1 6 2 5 32 125 Warm Up Multiply. Write each fraction in simplest form. 1. 2. Write each fraction as a decimal. 3. 4. 6 25 1 8 0.4 0.256

**Learn to find the probabilities of independent and dependent** events.

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

**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.

**Example 2** 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.

**12** 12 12 12 18 In each box, P(blue) = . · · = = Example 3: 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. 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

**In each box, P(green) = .** 1 2 12 12 12 12 18 In each box, P(blue) = . · · = = Example 4: Finding the Probability of Independent Events 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

**14** 14 14 In each box, P(blue) = . 1 16 · = = Example 5 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 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

**14** 14 14 14 In each box, P(blue) = . In each box, P(red) = . 1 16 · = = Example 6 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? Multiply. P(blue, red) = 0.0625

**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.

**23** 69 = Example 7: Find the Probability of Dependent Events 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? 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. P(first consonant) =

**58** 58 23 5 12 · = The probability of choosing two letters that are both consonants is . 5 12 Continue Example 7 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.

**There are 6 red gumballs, 4 yellow, 8 green, and 2 blue. ** Once one is picked, it is NOT replaced. What is the probability you will draw a blue first, then red, then yellow?

**Remember!** Two mutually exclusive events cannot both happen at the same time.

**Lesson Quizzes** Standard Lesson Quiz Lesson Quiz for Student Response Systems

**5** 33 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

**Lesson Quiz for Student Response Systems** 1. Identify the dependent event. A. drawing an apple from a basket and then drawing an orange without replacing the apple B. drawing an apple from a basket and then drawing an orange after replacing the apple C. getting a 3 and then a 2 when a fair number cube is rolled twice D. spinning an even number on a spinner two times in a row

**Lesson Quiz for Student Response Systems** 2. Identify the independent event. A. pulling out a white sock and then a blue sock from a dresser B. choosing a student from 7th Grade and a student from 8th Grade C. drawing a six and then a diamond from a deck of well shuffled cards D. choosing three members from a club

**Lesson Quiz for Student Response Systems** 3. The letters in the word PROBABILITY are placed in a box. If two cards are chosen at random, what is the probability that they will both have the letter B? A. B. C. D.