1. Pre-Algebra Independent and Dependent Events 9.6

2. Evaluate 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 10! 7! Warm Up

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

4. independent events dependent events Vocabulary

5. 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.

6. 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: Classifying Events as Independent or Dependent

7. 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. Try This

9. 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. P(blue, blue, blue) = In each box, P(blue) =. 1212 1212 · 1212 · 1212 = 1818 = 0.125 Multiply. Example: Finding the Probability of Independent Events

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

11. C. What is the probability of choosing at least one blue marble? 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. Example: Finding the Probability of Independent Events

12. 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. In each box, P(blue) =. 1414 P(blue, blue) = 1414 · 1414 = 1 16 = 0.0625 Multiply. Try This

13. 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? In each box, P(blue) =. 1414 P(blue, red) = 1414 · 1414 = 1 16 = 0.0625 Multiply. In each box, P(red) =. 1414 Try This

14. 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? 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. Try This

15. 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.

16. 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) = 2323 6969 = Example Find the Probability of Dependent Events

17. 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. Example Continued

18. 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. 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 = Example: Find the Probability of Dependent Events

19. 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) Example Continued

20. 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) = 5959 Try This

21. 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 = Try This Continued

22. 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. 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 Try This

23. 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 Try This Continued

24. 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 Lesson Quiz