1. 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:

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

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

4. 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.

5. 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.

6. 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.

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

8. 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.

9. 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.

10. 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 – = 0.875

11. 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.

12. 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.

13. 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 – =

14. 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

15. Homework Solution lesson 10.5
8) ) ) ) independent 16) dependent

16. Homework
Lesson 10.6 page 668 #9-16 ALL

17. Lesson 10.6 Dependent Events

18. 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

19. 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) =

20. 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

21. 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) =

22. 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

23. 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.

24. 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

25. 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

26. 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 )= =
𝟐𝟒 𝟗𝟏
P(black, blue) =
Slide 26

27. 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 = = 5
# of socks after 1 blue is removed
Total # of socks after 1 blue is removed

28. 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)

29. 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