Homework 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.
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Homework
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?
independent
dependent
533
Warm UpMultiply. Write each fraction in simplest form.
1. 2.
Write each fraction as a decimal.
3. 4.
6 25
1 8
0.4
3 4
2 5
3 5
1 6
2 5
32 125
0.256
Vocabulary
compound events
independent eventsdependent events
A compound event is made up of two 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.
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
Additional Example 1: Classifying Events as Independent or Dependent
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.
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
Check It Out! Example 1
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.
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?
Additional Example 2: Finding the Probability of Independent Events
The outcome of each choice does not affect the outcome of the other choices, so the choices are independent.
P(blue, blue, blue) =
In each box, P(blue) = .12
12
· 12
· 12
= 18
= 0.125 Multiply.
B. What is the probability of choosing a blue marble, then a green marble, and then a blue marble?
Additional Example 2: Finding the Probability of Independent Events
P(blue, green, blue) = 12
· 12
· 12
= 18
= 0.125 Multiply.
In each box, P(blue) = .12
In each box, P(green) = .1 2
C. What is the probability of choosing at least one blue marble?
Additional Example 2: Finding the Probability of Independent Events
1 – 0.125 = 0.875
Subtract from 1 to find 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 2P(not blue, not blue, not blue) =
12
· 12
· 12
= 18
= 0.125 Multiply.
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?
Check It Out! Example 2
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
P(blue, blue) = 14
· 14
= 116
= 0.0625 Multiply.
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?
Check It Out! Example 2
In each box, P(blue) = .14
P(blue, red) = 14
· 14
= 116
= 0.0625 Multiply.
In each box, P(red) = .14
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?
Check It Out! Example 2
In each box, P(blue) = .14
P(not blue, not blue) = 34
· 34
= 916
= 0.5625 Multiply.
Think: P(at least one blue) + P(not blue, not blue) = 1.
1 – 0.5625 = 0.4375Subtract from 1 to find the probability of choosing at least one blue marble.
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.
The letters in the word dependent are placed in a box.
A. If two letters are chosen at random, without replacing the first letter, what is the probability that they will both be consonants?
Additional Example 3: Find the Probability of Dependent Events
P(first consonant) = 23
69
=
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.
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.
P(second consonant) = 58
5 12
58
23
· =
The probability of choosing two letters that are both consonants is .5
12
Multiply.
B. If two letters are chosen at random, what is the probability that they will both be
consonants or both be vowels?
Additional Example 3: Find the Probability of Dependent Events
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.
If the first letter chosen was a vowel, there are now only 2 vowels and 8 total letters left in the box.
P(first vowel) = 13
39
=
Additional Example 3B Continued
Find the probability that the second letter chosen is a vowel.
The events of both consonants and both vowels are mutually exclusive, so you can add their probabilities.
P(second vowel) = 14
28
=
1 12
14
13
· = Multiply.
12
5 12
1 12
+ = 6 12
=
The probability of getting two letters that are
either both consonants or both vowels is .12
P(consonant) + P(vowel)
Two mutually exclusive events cannot both happen at the same time.
Remember!
The letters in the phrase I Love Math are placed in a box.
A. If two letters are chosen at random, without replacing the first letter, what is the probability that they will both be consonants?
Check It Out! Example 3
P(first consonant) = 59
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.
Check It Out! Example 3A Continued
P(second consonant) =
5 18
12
59
· =
The probability of choosing two letters that are both consonants is .5
18
Multiply.
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.
12
48
=
B. If two letters are chosen at random, what is the probability that they will both be consonants or both be vowels?
Check It Out! Example 3
There are two possibilities: 2 consonants or 2 vowels. The probability of 2 consonants was calculated in Check It Out 3A. Now find the probability of getting 2 vowels.
Find the probability that the first letter chosen is a vowel.
If the first letter chosen was a vowel, there are now only 3 vowels and 8 total letters left in the box.
P(first vowel) = 49
Check It Out! Example 3B Continued
Find the probability that the second letter chosen is a vowel.
The events of both consonants and both vowels are mutually exclusive, so you can add their probabilities.
P(second vowel) = 38
12 72
38
49
· = Multiply.16
=
49
5 18
1 6
+ = 8 18
= P(consonant) + P(vowel)
The probability of getting two letters that are
either both consonants or both vowels is .49
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?
independent
dependent
533
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