Independent and Dependent Probability

1. Independent and Dependent Probability

2. A compound eventis 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 eventsif the occurrence of one event does not affect the probability of the other. Events are dependent eventsif the occurrence of one does affect the probability of the other.

3. Example 1: • Determine if the events are dependent or independent. • 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.

4. Example 2: 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.

5. 12 12 12 12 18 In each box, P(blue) = · · = = Three separate boxes each have one blue marble and one green marble. One marble is chosen from each box. What is the P(blue) 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

6. In each box, P(green) = 1 2 18 12 12 12 12 In each box, P(blue) = · · = = Independent 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

7. Dependent 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.

8. 23 69 = Dependent 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) =

9. 58 58 23 5 12 · = Theprobability of choosing two letters that are both consonants is 512 Dependent 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.

10. 13 39 = Dependent 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.

11. 12 13 14 28 14 12 = 1 12 5 12 6 12 1 12 · = = = + The probability of getting two letters that are either both consonants or both vowels is Dependent 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)