1. Pre-Algebra 9-7 Independent and Dependent Events Learn to find the probabilities of independent and dependent events.

2. Pre-Algebra 9-7 Independent and Dependent Events The probability of an event, written P (event), is a number from 0 (or 0%) to 1 (or 100%) that tells you how likely the event is to happen. A probability of 0 means the event is impossible, or can never happen. A probability of 1 means the event is certain, or has to happen. The probabilities of all the outcomes in the sample space add up to 1.

3. Pre-Algebra 9-7 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.

4. Pre-Algebra 9-7 Independent and Dependent Events 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 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.

5. Pre-Algebra 9-7 Independent and Dependent Events 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 Try This Together: 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.

6. Pre-Algebra 9-7 Independent and Dependent Events

7. Pre-Algebra 9-7 Independent and Dependent Events 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? 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 · · = 18 = 0.125 Multiply.

8. Pre-Algebra 9-7 Independent and Dependent Events B. What is the probability of choosing a blue marble, then a green marble, and then a blue marble? Example 2B: Finding the Probability of Independent Events P (blue, green, blue) = 12 · · = 18 = 0.125 Multiply. In each box, P (blue) =. 12 In each box, P (green) =. 1212

9. Pre-Algebra 9-7 Independent and Dependent Events C. What is the probability of choosing at least one blue marble? 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) =. 1212 P (not blue, not blue, not blue) = 12 · · = 18 = 0.125 Multiply.

10. Pre-Algebra 9-7 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.

11. Pre-Algebra 9-7 Independent and Dependent Events

12. Pre-Algebra 9-7 Independent and Dependent Events 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? Try This Together: Example 2A The outcome of each choice does not affect the outcome of the other choices, so the choices are independent. In each box, P (blue) =. 14 P (blue, blue) = 14 · = 1 16 = 0.0625 Multiply.

13. Pre-Algebra 9-7 Independent and Dependent Events 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? Try This: Example 2B In each box, P (blue) =. 14 P (blue, red) = 14 · = 1 16 = 0.0625 Multiply. In each box, P (red) =. 14

14. Pre-Algebra 9-7 Independent and Dependent Events 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? Try This: Example 2C In each box, P (blue) =. 14 P (not blue, not blue) = 34 · = 9 16 = 0.5625 Multiply. 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.

15. Pre-Algebra 9-7 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.

16. Pre-Algebra 9-7 Independent and Dependent Events 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? Example 3A: Find the Probability of Dependent Events P (first consonant) = 2323 6969 =

17. Pre-Algebra 9-7 Independent and Dependent Events 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 10 24 = The probability of choosing two letters that are both consonants is. 5 12 Multiply. 5 12 5858 2323 · =

18. Pre-Algebra 9-7 Independent and Dependent Events B. If two letters are chosen at random, what is the probability that they will both be consonants or both be vowels? Example 3B: 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 =

19. Pre-Algebra 9-7 Independent and Dependent Events 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)

20. Pre-Algebra 9-7 Independent and Dependent Events 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? Try This: Example 3A P (first consonant) = 5959

21. Pre-Algebra 9-7 Independent and Dependent Events Try This: 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. Pre-Algebra 9-7 Independent and Dependent Events B. If two letters are chosen at random, what is the probability that they will both be consonants or both be vowels? Try This: Example 3B 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

23. Pre-Algebra 9-7 Independent and Dependent Events Try This: 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 1 6 + = 8 18 = P(consonant) + P(vowel) The probability of getting two letters that are either both consonants or both vowels is. 4949

24. Pre-Algebra 9-7 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.

25. Pre-Algebra 9-7 Independent and Dependent Events

26. Pre-Algebra 9-7 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.