Understanding Independent and Dependent Events in Probability

1. Independent and Dependent Events 10-5 Course 3 Warm Up Problem of the Day Lesson Presentation

2. Independent and Dependent Events 10-5 3 4 2 5 3 5 1 6 2 5 32 125 Course 3 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

3. Independent and Dependent Events 10-5 red blue Course 3 Problem of the Day The area of a spinner is 75% red and 25% blue. However, the probability of its landing on red is only 50%. Sketch a spinner to show how this can be. Possible answer:

4. Independent and Dependent Events 10-5 Course 3 Learn to find the probabilities of independent and dependent events.

5. Independent and Dependent Events 10-5 Course 3 Insert Lesson Title Here Vocabulary compound events independent events dependent events

6. Independent and Dependent Events 10-5 Course 3 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. 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.

7. Independent and Dependent Events 10-5 Course 3 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.

8. Independent and Dependent Events 10-5 Course 3 Check It Out: 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.

9. Independent and Dependent Events 10-5 Course 3

10. Independent and Dependent Events 10-5 18 12 12 12 12 In each box, P(blue) = . · · = = Course 3 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. 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

11. Independent and Dependent Events 10-5 In each box, P(green) = . 1 2 12 12 18 12 12 In each box, P(blue) = . · · = = Course 3 Additional Example 2B: 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

12. Independent and Dependent Events 10-5 In each box, P(not blue) = . 18 12 12 12 1 2 · · = = Course 3 Additional Example 2C: Finding the Probability of Independent Events 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

13. Independent and Dependent Events 10-5 14 14 14 In each box, P(blue) = . 1 16 · = = Course 3 Check It Out: Example 2A 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. Independent and Dependent Events 10-5 14 14 14 14 In each box, P(blue) = . In each box, P(red) = . 1 16 · = = Course 3 Check It Out: Example 2B 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

15. Independent and Dependent Events 10-5 14 34 34 In each box, P(blue) = . 9 16 · = = Course 3 Check It Out: Example 2C 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? 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

16. Independent and Dependent Events 10-5 Course 3 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.

17. Independent and Dependent Events 10-5 23 69 = Course 3 Additional Example 3A: 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) =

18. Independent and Dependent Events 10-5 58 58 23 5 12 · = The probability of choosing two letters that are both consonants is . 5 12 Course 3 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.

19. Independent and Dependent Events 10-5 13 39 = Course 3 Additional Example 3B: Find the Probability of Dependent Events 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.

20. Independent and Dependent Events 10-5 13 12 12 14 28 14 = 5 12 6 12 1 12 1 12 · = = = + The probability of getting two letters that are either both consonants or both vowels is . Course 3 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)

21. Independent and Dependent Events 10-5 Remember! Two mutually exclusive events cannot both happen at the same time. Course 3

22. Independent and Dependent Events 10-5 59 Course 3 Check It Out: Example 3A The letters in the phrase I Love Math 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 dependant. Find the probability that the first letter chosen is a consonant. P(first consonant) =

23. Independent and Dependent Events 10-5 12 48 12 59 = 5 18 · = The probability of choosing two letters that are both consonants is . 5 18 Course 3 Check It Out: 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.

24. Independent and Dependent Events 10-5 49 Course 3 Check It Out: Example 3B 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.

25. Independent and Dependent Events 10-5 = 49 49 38 38 49 16 5 18 12 72 8 18 1 6 · = = = + The probability of getting two letters that are either both consonants or both vowels is . Course 3 Check It Out: 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)

26. Independent and Dependent Events 10-5 5 33 Course 3 Insert Lesson Title Here 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