Learn to find the probabilities of independent and dependent events.

Vocabulary independent events 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 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.

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.

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 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 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

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.

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 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

Additional Example 3B: Find the Probability of Dependent Events 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. 39 = 13 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.

Additional Example 3B Continued Find the probability that the second letter chosen is a vowel. 28 = 14 P(second vowel) = 14 13 · = 1 12 Multiply. The events of both consonants and both vowels are mutually exclusive, so you can add their probabilities. 5 12 1 12 + = 6 12 = 12 P(consonant) + P(vowel) The probability of getting two letters that are either both consonants or both vowels is . 12

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 5 33