Independent and Dependent Events Course 3 9-7 Independent and Dependent Events 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. red blue Possible answer:

Independent and Dependent Events Course 3 9-7 Independent and Dependent Events Learn to find the probabilities of independent and dependent events.

Insert Lesson Title Here Course 3 9-7 Independent and Dependent Events Insert Lesson Title Here Vocabulary independent events dependent events

Independent and Dependent Events Course 3 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.

Additional Example 1: Classifying Events as Independent or Dependent Course 3 9-7 Independent and Dependent Events 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.

Independent and Dependent Events Course 3 9-7 Independent and Dependent Events Try This: 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.

Independent and Dependent Events Course 3 9-7 Independent and Dependent Events

Additional Example 2A: Finding the Probability of Independent Events Course 3 9-7 Independent and Dependent Events 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. 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) = . 12 12 · = 18 = P(blue, blue, blue) = 0.125 Multiply.

Additional Example 2B: Finding the Probability of Independent Events Course 3 9-7 Independent and Dependent Events Additional Example 2B: Finding the Probability of Independent Events B. What is the probability of choosing a blue marble, then a green marble, and then a blue marble? In each box, P(blue) = . 12 In each box, P(green) = . 1 2 12 · = 18 = P(blue, green, blue) = 0.125 Multiply.

Additional Example 2C: Finding the Probability of Independent Events Course 3 9-7 Independent and Dependent Events Additional Example 2C: Finding the Probability of Independent Events C. 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. In each box, P(not blue) = . 1 2 P(not blue, not blue, not blue) = 12 · = 18 = 0.125 Multiply. Subtract from 1 to find the probability of choosing at least one blue marble. 1 – 0.125 = 0.875

Independent and Dependent Events Course 3 9-7 Independent and Dependent Events Try This: Example 2A 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) = . 14 14 · = 1 16 = P(blue, blue) = 0.0625 Multiply.

Independent and Dependent Events Course 3 9-7 Independent and Dependent Events Try This: Example 2B 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) = . 14 In each box, P(red) = . 14 14 · = 1 16 = P(blue, red) = 0.0625 Multiply.

Independent and Dependent Events Course 3 9-7 Independent and Dependent Events Try This: Example 2C 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? Think: P(at least one blue) + P(not blue, not blue) = 1. In each box, P(blue) = . 14 34 · = 9 16 = 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

Warm Up #25 Independent and Dependent Events 9-7 1. 8! 2. 40,320 Course 3 9-7 Independent and Dependent Events Warm Up #25 Evaluate each expression. 1. 8! 2. 3. 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? 40,320 10!7! 720 1/8

Homework Solution lesson 10.5 8) 0.375 10) 0.075 12) 0.025 14) independent 16) dependent

Homework Lesson 10.6 page 668 #9-16 ALL

Lesson 10.6 Dependent Events

Dependent Event What happens during the second event depends upon what happened before. In other words, the result of the second event will change because of what happened first. In plain English this means that after event A occurs you reduce the # of favorable outcomes and the # of total outcomes Slide 18

Additional Example 3A: Find the Probability of Dependent Events Course 3 9-7 Independent and Dependent Events Additional Example 3A: Find the Probability of 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? 69 = 23 P(first consonant) =

Additional Example 3A Continued Course 3 9-7 Independent and Dependent Events 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. 58 P(second consonant) = 58 23 · = 5 12 Multiply. The probability of choosing two letters that are both consonants is . 5 12

Independent and Dependent Events Course 3 9-7 Independent and Dependent Events Try This: Example 3A 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? 59 P(first consonant) =

Try This: Example 3A Continued Course 3 9-7 Independent and Dependent Events Try This: 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. 48 = 12 P(second consonant) = 12 59 · = 5 18 Multiply. The probability of choosing two letters that are both consonants is . 5 18

Independent and Dependent Events Course 3 9-7 Independent and Dependent Events Try This: Example 3B 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. 49 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.

Try This: Example 3B Continued Course 3 9-7 Independent and Dependent Events Try This: Example 3B Continued 38 Find the probability that the second letter chosen is a vowel. P(second vowel) = 38 49 · = 12 72 16 = Multiply. The events of both consonants and both vowels are mutually exclusive, so you can add their probabilities. 5 18 1 6 + = 8 18 = 49 P(consonant) + P(vowel) The probability of getting two letters that are either both consonants or both vowels is . 49

Dependent Event Example: There are 6 black pens and 8 blue pens in a jar. If you take a pen without looking and then take another pen without replacing the first, what is the probability that you will get 2 black pens? P(black first) = P(black second) = (There are 13 pens left and 5 are black) THEREFORE……………………………………………… P(black, black) = Slide 25

Dependent Event ( 6 14 ) 48 182 = ( 8 13 )= Example: There are 6 black pens and 8 blue pens in a jar. If you take a pen without looking and then take another pen without replacing the first, what is the probability that you will get a black and a blue? P(black first) = P(blue second) = 8 13 (There are 13 pens left and 5 are black) THEREFORE……………………………………………… ( 6 14 ) ( 8 13 )= 48 182 = 𝟐𝟒 𝟗𝟏 P(black, blue) = Slide 26

Example There are 4 red, 8 yellow and 6 blue socks in a drawer. Once a sock is selected it is not replaced. Find the probability that 2 blue socks are chosen. P(1st blue sock) = 6 18 P(2nd blue sock) = 5 17 P(Two blue socks) = 6 ∙ 5 = 30 = 5 18 17 306 51 # of socks after 1 blue is removed Total # of socks after 1 blue is removed

Your Turn In the notes section of your notebook write a probability statement and determine the probability There are 3 yellow, 5 red, 4 blue and 8 green candies in a bag. Once a candy is removed it is not replaced. Find the probability: P(two red candies) P(two blue candies) P(yellow candy followed by a blue candy)

WARM-UP #26 Independent and Dependent Events 9-7 Course 3 9-7 Independent and Dependent Events Insert Lesson Title Here WARM-UP #26 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 5 33