Probability of Independent and
10-6 Dependent Events
Learn to find the probability of
independent and dependent events.
Probability of Independent and
10-6 Dependent Events
Raji and Kara must each choose a solo from a
list of music pieces to play for their recital.
If Raji’s choice has no effect on Kara’s choice
and vice versa, the events are independent.
For independent events, the occurrence of
one event has no effect on the probability
that a second event will occur.
Probability of Independent and
10-6 Dependent Events
If once Raji chooses a solo, Kara must choose
from the remaining solos, 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
10-6 Dependent Events
Example A
Decide whether the set of events are dependent
or independent. Explain your answer.
1. Erika rolls a 3 on one number cube and a 2 on
another number cube.
Independent – rolling the first cube does not
affect the outcome of rolling the second cube
2. Tomoko chooses a seventh-grader for her term
from a group of seventh- and eighth-graders, and
then Juan chooses a different seventh-grader from
the remaining students.
Dependent – Tomoko’s first choice affects the
outcome of Juan’s choice
Probability of Independent and
10-6 Dependent Events
To find the probability that two
independent events will happen, multiply
the probabilities of the two 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
Probability of Independent and
10-6 Dependent Events
Example B
Find the probability of flipping a coin and
getting heads and then rolling a 6 on a
number cube.
The outcome of flipping a coin does not affect the
outcome of rolling a number cube so the events
are independent.
P(heads and 6) = P(heads) · P(6)
= 1 · 1 =1
2 6
12
The probability of flipping a· heads and rolling a 6 is 1
12
Probability of Independent and
10-6 Dependent Events
YOU TRY
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
Probability of Independent and
10-6 Dependent Events
YOU TRY
Find the probability of rolling a six on the first
roll of a 1-6 number cube and rolling an odd
number of the second roll of the same cube.
The outcome of rolling a number cube does not
affect the outcome of rolling it again so the events
are independent.
P(6 and odd) = P(6) · P(odd)
= 1· 1
6 2
1
·
The probability of rolling a 6 and then rolling an odd is 12
Probability of Independent and
10-6 Dependent Events
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) • P(B after A)
Probability of Probability of
first event
both events
Probability of
second event
Probability of Independent and
10-6 Dependent Events
Example C
You are randomly choosing 2 cards from a
regular deck of 52 cards. What is the
probability you will pick an Ace and then a
Jack without replacing the first card?
Pulling the first card changes the number of cards
left, so the events are dependent.
Probability of Independent and
10-6 Dependent Events
Example C continued
1 There are 4 aces out of 52 cards.
4
P(Ace) = 52 = 13
There are 4 jacks out of 51 cards.
4
P(Jack) = 51
(since one ace is already removed)
P(Ace and then Jack) = P(A) · P(B after A)
1 · 4
= 13
51
=
4 Multiply.
663
The probability of choosing an ace and then a jack is
4·
663
Probability of Independent and
10-6 Dependent Events
YOU Try
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?
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
10-6 Dependent Events
You try Continued
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)
· 3
= 5
8 7
= 15 Multiply.
56
The probability of Juan choosing a historical book and
then choosing a science-fiction book is 15·
56
Probability of Independent and
10-6 Dependent Events
You try
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.
Probability of Independent and
10-6 Dependent Events
You try 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)
· 3
= 4
7 6
Multiply.
= 12 or 2
7
42
The probability of Alice selecting a black card and
then choosing a red card is 2 .
7
Probability
of Independent
and
Independent
and
10-6 Dependent
10.6
DependentEvents
Events
Extra Example
The letters in the word dependent are placed
in a box.
A. If two letters are chosen at random and not
replaced, what is the probability that they will
both be consonants?
P(first consonant) = 6 = 2
9
3
P(second consonant) = 5
8
2
5
10
5
=
=
3
8
24
12
Pre-Algebra
Probability of Independent and
10-6 Dependent Events
Lesson Quiz: Part I
Decide whether each event is independent
or dependent. Explain.
1. Mary chooses a game piece from a board
game, and then Jason chooses a game piece from
three remaining pieces. Dependent; Jason has
fewer pieces from which to choose.
2. Find the probability of spinning an evenly
divided spinner numbered 1–8 and getting a
composite number on one spin and getting an
odd number on a second spin.
3
16
Probability of Independent and
10-6 Dependent Events
Lesson Quiz: Part II
3. Sarah picks 2 hats at random from 5 bill caps
and 3 beanies. What is the probability that
both are bill caps?
5
14