Download Independent and Dependent Events

Raji and Kara must each choose a topic from a list of
topics to research for their class. If Raji’s choice has
no effect on Kara’s choice and vice versa, the events
are independent.
Independent and Dependent
Events
For independent events, the occurrence of one
event has no effect on the probability that a second
event will occur.
If once Raji chooses a topic, Kara must choose from
the remaining topics, then the events are dependent.
For dependent events, the occurrence of one event
does have an effect on the probability that a second
event will occur.
Probability of Independent and
Dependent Events
Additional Example 1A: Determining Whether Events
Are Independent or Dependent
Additional Example 1B: Determining Whether Events
Are Independent or Dependent
Decide whether the set of events are dependent
or independent. Explain your answer.
Decide whether the set of events are dependent
or independent. Explain your answer.
A. Kathi draws a 4 from a set of cards numbered
1–10 and rolls a 2 on a number cube.
B. Yuki chooses a book from the shelf to read,
and then Janette chooses a book from the
books that remain.
Since Janette cannot pick the same book that
Yuki picked, and since there are fewer books
for Janette to choose from after Yuki chooses,
the events are dependent (no
replacement).
Since the outcome of drawing the card does not
affect the outcome of rolling the cube, the
events are independent.
Course 2
Insert Lesson Title Here
Insert Lesson Title Here
Try This: Example 1A
Try This: Example 1B
Decide whether the set of events are dependent
or independent. Explain your answer.
Decide whether the set of events are dependent
or independent. Explain your answer.
A. Joann flips a coin and gets a head. Then she
rolls a 6 on a number cube.
Since flipping the coin does not affect the
outcome of rolling the number cube, the events
are independent.
B. Annabelle chooses a blue marble from a set
of three, each of different colors, and
then Louise chooses a second marble from the
remaining two marbles.
Since they are picking from the same set of three
marbles, they cannot pick the same color marble.
The events are dependent (no replacement).
1
Additional Example 1: Classifying Events as
Independent or Dependent
Try This: Example 1
Determine if the events are dependent or
independent.
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.
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.
To find the probability that two independent
events will happen, multiply the probabilities
of the two events.
Additional Example 2: Finding the Probability of
Independent Events
Probability of Two Independent Events
P(A and B) = P(A) • P(B)
Probability of
both events
Probability of
first event
Probability of
second event
Find the probability of choosing a green
marble at random from a bag containing 5
green and 10 white marbles and then flipping
a coin and getting tails.
The outcome of choosing the marble does not
affect the outcome of flipping the coin, so the
events are independent.
P(green and tails) = P(green) · P(tails)
=1· 1
3 2
The probability of choosing a green marble and a
coin landing on tails is 1
6·
Course 2
Probability of Independent and
Insert
Lesson
Title Here
Dependent
Events
Try This: Example 2
Find the probability of choosing a red marble
at random from a bag containing 5 red and 5
white marbles and then flipping a coin and
getting heads.
The outcome of choosing the marble does not affect
the outcome of flipping the coin, so
the events are independent.
P(red and heads) = P(red) · P(heads)
= 1· 1
2 2
The probability of choosing a red marble and a
coin landing on heads is 1 ·
4
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.
1
In each box, P(blue) =
.
2
1
P(blue, blue, blue) = 1 ·
· 12 = 18 = 0.125 Multiply.
2
2
Course 2
2
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?
1
.
2
1
In each box, P(green) =
.
2
In each box, P(blue) =
P(blue, green, blue) = 1
2
·
1
2
·
1 = 1 = 0.125 Multiply.
8
2
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.
1
In each box, P(not blue) =
.
2
P(not blue, not blue, not blue) =
1
· 12 · 12 = 18 = 0.125 Multiply.
2
Subtract from 1 to find the probability of
choosing at least one blue marble.
1 – 0.125 = 0.875
Try This: Example 2B
Try This: Example 2A
Two boxes each contain 4 marbles: red, blue,
green, and black. One marble is chosen from
each box.
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.
1
In each box, P(blue) =
.
4
B. 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, blue) = 1
4
·
1 = 1 = 0.0625
16
4
Multiply.
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.
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
P(blue, red) =
1
4
·
1 = 1 = 0.0625
16
4
Multiply.
To find the probability of two dependent
events, you must determine the effect
that the first event has on the probability
of the second event.
Probability of Two Dependent Events
P(A and B) = P(A)
•
Probability of Probability of
both events
first event
P(B after A)
Probability of
second event
3
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.
Additional Example 3: Finding the Probability of
Dependent Events
A reading list contains 5 historical books and 3
science-fiction books. What is the probability
that Juan will randomly choose a historical
book for his first report and a science-fiction
book for his second?
3. Multiply the probabilities.
The first choice changes the number of books left,
and may change the number of science-fiction
books left, so the events are dependent.
Probability of Independent and
Dependent Events
Insert Lesson Title Here
Additional Example 3 Continued
Try This: Example 3
5
P(historical) = 8 There are 5 historical books out of 8 books.
There are 3 science-fiction books left
P(science-fiction) = 3
7 out of 7 books.
P(historical and then science-fiction) = P(A) · P(B after A)
5 3
= 8· 7
= 15 Multiply.
56
The probability of Juan choosing a historical book and
then choosing a science-fiction book is15·
56
Alice was dealt a hand of cards consisting of 4
black and 3 red cards. Without seeing the
cards, what is the probability that the first
card will be black and the second card will be
red?
The first choice changes the total number of cards
left, and may change the number of red cards left,
so the events are dependent.
Course 2
Insert Lesson Title Here
Try This: Example 3 Continued
P(black) = 4
7
P(red) = 3
6
There are 4 black cards out of 7 cards.
There are 3 red cards left out of
6 cards.
P(black and then red card) = P(A) · P(B after A)
4 3
= 7· 6
Multiply.
= 12 or 2
7
42
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?
P(first consonant) = 6 = 2
3
9
The probability of Alice selecting a black card and
then choosing a red card is 2 .
7
4
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) = 5
8
5
2
5
·
=
12
3
8
Multiply.
The probability of choosing two letters that are
both consonants is 5 .
12
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.
3
12
4
The events of both consonants and both vowels are
mutually exclusive, so you can add their probabilities.
1
5
1
6
=
=
+
2
12 12 12
Additional Example 3B: Find the Probability of
Dependent Events
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 Example 3A. Now find the probability
of getting 2 vowels.
Find the probability that
1 the first letter chosen is a
P(first vowel) = 3 =
3 vowel.
9
If the first letter chosen was a vowel, there are
now only 2 vowels and 8 total letters left in the
box.
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?
P(first consonant) = 5
9
P(consonant) + P(vowel)
The probability of getting two letters that are
either both consonants or both vowels is 1 .
2
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.
1
P(second consonant) = 4 =
2
8
5
5
1
·
=
18
9
2
Multiply.
The probability of choosing two letters that are
both consonants is 5 .
18
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
P(first vowel) = 4
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.
5
Try This: 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.
4
5
1 = 8
=
+
9
18
18
6
P(consonant) + P(vowel)
The probability of getting two letters that are
either both consonants or both vowels is 4 .
9
6