1. Independent and Dependent Events
Course 3
10-5
Independent and Dependent Events
EQ: How do I find the probabilities of independent and dependent events?

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

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

4. Additional Example 1: Classifying Events as Independent or Dependent
Course 3
10-5
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.

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

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

7. Additional Example 2A: Finding the Probability of Independent Events
Course 3
10-5
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.
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.

8. Additional Example 2B: Finding the Probability of Independent Events
Course 3
10-5
Independent and Dependent Events
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?
In each box, P(blue) = . 12
In each box, P(green) = .
1 2 12 = 18 =
P(blue, green, blue) =
0.125
Multiply.

9. Additional Example 2C: Finding the Probability of Independent Events
Course 3
10-5
Independent and Dependent Events
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.
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 – = 0.875

10. Independent and Dependent Events
Course 3
10-5
Independent and Dependent Events
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.
In each box, P(blue) = . 14 14 = 1 16 =
P(blue, blue) =
0.0625
Multiply.

11. Independent and Dependent Events
Course 3
10-5
Independent and Dependent Events
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?
In each box, P(blue) = . 14
In each box, P(red) = 14 14 = 1 16 =
P(blue, red) =
0.0625
Multiply.

12. Independent and Dependent Events
Course 3
10-5
Independent and Dependent Events
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.
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 – =

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

14. Additional Example 3A: Find the Probability of Dependent Events
Course 3
10-5
Independent and Dependent Events
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. 69 = 23
P(first consonant) =

15. Additional Example 3A Continued
Course 3
10-5
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

16. Additional Example 3B: Find the Probability of Dependent Events
Course 3
10-5
Independent and Dependent Events
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. 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.

17. Additional Example 3B Continued
Course 3
10-5
Independent and Dependent Events
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

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

19. Independent and Dependent Events
Course 3
10-5
Independent and Dependent Events
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. 59
P(first consonant) =

20. Check It Out: Example 3A Continued
Course 3
10-5
Independent and Dependent Events
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. 48 = 12
P(second consonant) = 12 59 =
5 18
Multiply.
The probability of choosing two letters that are both consonants is
5 18

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

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

23. Independent and Dependent Events Insert Lesson Title Here
Course 3
10-5
Independent and Dependent Events
Insert Lesson Title Here
You try
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