Download 10-5 Independent and Dependent Events

Independent and
10-5 Dependent Events
Warm Up
Problem of the Day
Lesson Presentation
Course 3
Independent
and
10-5 Dependent Events
Warm Up
Multiply. Write each fraction in simplest form.
1. 2  3
5
5
6
25
2. 1  3
6
4
1
8
Write each fraction as a decimal.
2
3.
5
Course 3
0.4
32
4.
125
0.256
Independent
and
10-5 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.
Possible answer:
Course 3
red
blue
Independent
and
10-5 Dependent Events
Learn to find the probabilities of
independent and dependent events.
Course 3
Independent
and
10-5 Dependent
Insert Lesson
Title Here
Events
Vocabulary
compound events
independent events
dependent events
Course 3
Independent
and
10-5 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.
Course 3
Independent
and
10-5 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
Tossing a coin does not affect rolling a number
cube, so the two events are independent.
B. getting 2 red gumballs out of a gumball
machine
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.
Course 3
Independent
and
10-5 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
The first roll of the number cube does not affect
the second roll, so the events are independent.
B. a computer randomly generating two of the
same numbers in a row
The first randomly generated number does not
affect the second randomly generated number, so
the two events are independent.
Course 3
Independent
and
10-5 Dependent Events
Course 3
Independent
and
10-5 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.
1
In each box, P(blue) =
.
2
P(blue, blue, blue) = 1 · 1 · 1 = 1 = 0.125 Multiply.
2
2
2
8
Course 3
Independent
and
10-5 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?
1
In each box, P(blue) =
.
2
1
In each box, P(green) =
.
2
P(blue, green, blue) = 1
2
Course 3
·
1
2
·
1 = 1 = 0.125 Multiply.
2
8
Independent
and
10-5 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.
1
In each box, P(not blue) =
.
2
P(not blue, not blue, not blue) =
1
1
1 = 1 = 0.125
Multiply.
·
·
2
2
2
8
Subtract from 1 to find the probability of
choosing at least one blue marble.
1 – 0.125 = 0.875
Course 3
Independent
and
10-5 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.
1
In each box, P(blue) =
.
4
P(blue, blue) = 1
4
Course 3
·
1 = 1 = 0.0625
4
16
Multiply.
Independent
and
10-5 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?
1
In each box, P(blue) =
.
4
1
In each box, P(red) =
.
4
P(blue, red) =
Course 3
1
4
·
1
1
=
= 0.0625
4
16
Multiply.
Independent
and
10-5 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.
1
In each box, P(blue) =
.
4
P(not blue, not blue) = 3 · 3 = 9 = 0.5625 Multiply.
4
4
16
Subtract from 1 to find the probability of choosing at
least one blue marble. 1 – 0.5625 = 0.4375
Course 3
Independent
and
10-5 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.
Course 3
Independent
and
10-5 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.
P(first consonant) = 6 = 2
9
3
Course 3
Independent
and
10-5 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.
5
P(second consonant) =
8
2 · 5 = 5
12
3
8
Multiply.
The probability of choosing two letters that are
both consonants is 5 .
12
Course 3
Independent
and
10-5 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
P(first vowel) = 3 = 1 the first letter chosen is a
9
3 vowel.
If the first letter chosen was a vowel, there are
now only 2 vowels and 8 total letters left in the
box.
Course 3
Independent
and
10-5 Dependent Events
Additional Example 3B Continued
P(second vowel) = 2 = 1
8
4
Find the probability that
the second letter chosen is
a vowel.
1 · 1 = 1
Multiply.
12
3
4
The events of both consonants and both vowels are
mutually exclusive, so you can add their probabilities.
5
1 = 6 = 1
+
2
12
12
12
P(consonant) + P(vowel)
The probability of getting two letters that are
either both consonants or both vowels is 1 .
2
Course 3
Independent
and
10-5 Dependent Events
Remember!
Two mutually exclusive events cannot
both happen at the same time.
Course 3
Independent
and
10-5 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.
P(first consonant) = 5
9
Course 3
Independent
and
10-5 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.
P(second consonant) = 4 = 1
8
2
5 · 1 = 5
18
9
2
Multiply.
The probability of choosing two letters that are
both consonants is 5 .
18
Course 3
Independent
and
10-5 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
4
P(first vowel) =
the first letter chosen is a
9
vowel.
If the first letter chosen was a vowel, there are
now only 3 vowels and 8 total letters left in the
box.
Course 3
Independent
and
10-5 Dependent Events
Check It Out: Example 3B Continued
P(second vowel) = 3
8
Find the probability that
the second letter chosen is
a vowel.
4 · 3 = 12 = 1
Multiply.
72
9
8
6
The events of both consonants and both vowels are
mutually exclusive, so you can add their probabilities.
5
+
18
1 = 8 = 4
9
18
6
P(consonant) + P(vowel)
The probability of getting two letters that are
either both consonants or both vowels is 4 .
9
Course 3
Independent
and
10-5 Dependent
Insert Lesson
Title Here
Events
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 dependent
2. spinning a 7 on a spinner three times in a row
independent
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? 5
33
Course 3