1. Homework 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 5 33

2. Warm Up Multiply. Write each fraction in simplest form. 1. 2.  Write each fraction as a decimal. 3.4. 6 25 1818 0.4 3434 2525 3535 1616 2525 32 125 0.256

3. Vocabulary compound events independent events dependent events

4. 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.

5. 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.

6. 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 Check It Out! 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.

8. 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 2: 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) =. 1212 1212 · 1212 · 1212 = 1818 = 0.125 Multiply.

9. B. What is the probability of choosing a blue marble, then a green marble, and then a blue marble? Additional Example 2: Finding the Probability of Independent Events P(blue, green, blue) = 1212 · 1212 · 1212 = 1818 = 0.125 Multiply. In each box, P(blue) =. 1212 In each box, P(green) =. 1212

10. C. What is the probability of choosing at least one blue marble? Additional Example 2: 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) =. 1212 P(not blue, not blue, not blue) = 1212 · 1212 · 1212 = 1818 = 0.125Multiply.

11. 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? Check It Out! Example 2 The outcome of each choice does not affect the outcome of the other choices, so the choices are independent. In each box, P(blue) =. 1414 P(blue, blue) = 1414 · 1414 = 1 16 = 0.0625 Multiply.

12. 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? Check It Out! Example 2 In each box, P(blue) =. 1414 P(blue, red) = 1414 · 1414 = 1 16 = 0.0625Multiply. In each box, P(red) =. 1414

13. 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? Check It Out! Example 2 In each box, P(blue) =. 1414 P(not blue, not blue) = 3434 · 3434 = 9 16 = 0.5625Multiply. Think: P(at least one blue) + P(not blue, not blue) = 1. 1 – 0.5625 = 0.4375 Subtract from 1 to find the probability of choosing at least one blue marble.

14. 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.

15. The letters in the word dependent are placed in a box. A. If two letters are chosen at random, without replacing the first letter, what is the probability that they will both be consonants? Additional Example 3: Find the Probability of Dependent Events P(first consonant) = 2323 6969 = 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.

16. 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) = 5858 5 12 5858 2323 ·= The probability of choosing two letters that are both consonants is. 5 12 Multiply.

17. B. If two letters are chosen at random, what is the probability that they will both be consonants or both be vowels? Additional Example 3: 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) = 1313 3939 =

18. 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) = 1414 2828 = 1 12 1414 1313 ·= Multiply. 1212 5 12 1 12 + = 6 12 = The probability of getting two letters that are either both consonants or both vowels is. 1212 P(consonant) + P(vowel)

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

20. The letters in the phrase I Love Math are placed in a box. A. If two letters are chosen at random, without replacing the first letter, what is the probability that they will both be consonants? Check It Out! Example 3 P(first consonant) = 5959 Because the first letter is not replaced, the sample space is different for the second letter, so the events are dependant. Find the probability that the first letter chosen is a consonant.

21. Check It Out! Example 3A Continued P(second consonant) = 5 18 1212 5959 ·= 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. 1212 4848 =

22. B. If two letters are chosen at random, what is the probability that they will both be consonants or both be vowels? Check It Out! Example 3 There are two possibilities: 2 consonants or 2 vowels. The probability of 2 consonants was calculated in Check It Out 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) = 4949

23. 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. P(second vowel) = 3838 12 72 3838 4949 ·= Multiply. 1616 = 4949 5 18 1616 + = 8 18 = P(consonant) + P(vowel) The probability of getting two letters that are either both consonants or both vowels is. 4949

24. 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 5 33