11.3
Independent Events
Lesson Objectives
Vocabulary
• Understand independent events.
• Use the multiplication rule and the addition rule of
probability to solve problems with independent events.
independent events
multiplication rule of probability
addition rule of probability
Understand Independent Events.
Suppose you are playing a game. You have a spinner with two congruent sections
and some color cards as shown below. Your goal is to randomly spin a 2 and draw
a red card.
1
B
R
Y
2
The event of spinning a 2 and the event of drawing a red card are considered
independent events. Two events are independent if the occurrence of one event
does not affect the probability of the other event. When you spin the spinner,
regardless of the result, it will not affect the probability of drawing a blue, yellow,
or red card.
Caution
Independent events are not the
same as mutually exclusive events.
Mutually exclusive events cannot
occur at the same time. But
independent events refer to whether
the occurrence of an event affects
the probability of the other event.
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Use the Multiplication Rule of Probability to Solve Problems with
Independent Events.
Consider the spinner and the 3 color cards again. You can draw a tree diagram to
represent the independent events that form the compound event and their
corresponding probabilities.
Spinner
Color Card
1
3
1
1
2
1
3
1
3
1
3
1
2
2
1
3
1
3
Outcome
B
(1, B)
R
(1, R)
Y
(1, Y)
B
(2, B)
R
(2, R)
Y
(2, Y)
B represents blue
R represents red
Y represents yellow
Because both events are independent, the outcome on the spinner does not affect
the probability of drawing a color card. The area of the spinner is equally divided
1
2
into 2 sections, so the probability of spinning one of the two numbers is .
Because there is an equal chance of drawing each color card, the probability of choosing
1
one of the three colors cards is . The probabilities are labeled on the branches of
3
the tree diagram.
You can see that there are a total of 6 equally likely outcomes, and spinning a 2 and
drawing a red card is one of those 6 equally likely outcomes. So, you can write the
probability of spinning a 2 and drawing a red card as follows:
P(2, R) 5
1
6
You can also use the multiplication rule of probability to find the probability of
spinning a 2 and drawing a red card.
P(2, R) 5 P(2) · P(R)
5
1 1
·
2 3
Multiply P(2) and P(R).
5
1
6
Simplify.
Continue on next page
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In general, for two independent events A and B, the multiplication rule of
probability states that:
P(A and B) 5 P(A) · P(B)
Suppose the blue card is replaced by another red card as shown.
1
R
R
Y
2
A tree diagram can be drawn to represent the possible outcomes as shown.
Spinner
Color Card
1
3
1
1
2
1
3
1
3
1
3
1
2
2
1
3
1
3
Outcome
R
(1, R)
R
(1, R)
Y
(1, Y)
R
(2, R)
R
(2, R)
Y
(2, Y)
R represents red
Y represents yellow
You can see that there is a total of 6 outcomes. To find the probability of spinning a 2
and drawing a red card, you can see that 2 out of 6 outcomes are favorable.
So, you can write the probability of spinning a 2 and drawing a red card as:
238
P(2, R) 5
2
6
5
1
3
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The tree diagram constructed previously for this compound event with two red cards
can be simplified by combining the identical outcomes (2, R) and (2, R) together as
shown below.
Spinner
Color Card
2
3
Outcome
R
(1, R)
Y
(1, Y)
R
(2, R)
1
1
2
1
3
1
2
2
3
2
1
3
Y
(2, Y)
R represents red
Y represents yellow
You can clearly see that the probability of drawing a red card is greater than in the
last example when drawing each color was equally likely. The simple event of drawing
a color card has become a biased event, because the probability of drawing a red
card is not the same as the probability of drawing a yellow card.
Using the multiplication rule to find the probability of spinning a 2 and drawing a red
card:
P(2, R) 5 P(2) · P(R)
5
1 2
·
2 3
Multiply P(2) and P(R).
5
1
3
Simplify.
Caution
For compound events involving
biased outcomes, the probability of
an outcome is not necessarily equal
to
1
Number of different outcomes
because the outcomes are not
equally likely.
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Example 6
Solve probability problems involving two independent events.
A game is played with a fair coin and a fair six-sided number die. To win the game,
you need to randomly obtain heads on a fair coin and a 3 on a fair number die.
a) Draw a tree diagram to represent this compound event.
Solution
Coin
Number Die
1
1
6
Outcome
(H, 1)
The events are
independent since
1
6
throwing a coin and
2
(H, 2)
a number die do not
affect the results of
1
6
(H, 3)
4
(H, 4)
5
(H, 5)
6
(H, 6)
1
(T, 1)
2
(T, 2)
3
(T, 3)
4
(T, 4)
5
(T, 5)
1
6
H
1
6
1
2
each other.
3
1
6
1
6
1
2
1
6
1
6
1
6
T
1
6
1
6
6
240
(T, 6)
H represents heads
T represents tails
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b)
Use the multiplication rule of probability to find the probability of winning the
game in one try.
Solution
Think Math
P(winning the game) 5 P(H, 3)
5 P(H) · P(3)
5
1 1
·
2 6
5
1
12
If the coin and number die are
biased, could the chance of winning
the game be the same? Explain.
The probability of winning the game in one try is
1
.
12
Guided Practice
Solve. Show your work.
1 A game is played with a bag of 6 color tokens and a bag of 6 letter tiles. The 6 tokens
consist of 2 green tokens, 1 yellow token, and 3 red tokens. The 6 letter tiles
consist of 4 tiles of letter A and 2 tiles of letter B. To win the game, you need to
get a yellow token and a tile of letter B from each bag.
a)
Copy and complete the tree diagram.
Token
Letter Tile
Outcome
4
6
A
( ? ,
? )
?
?
B
( ? ,
? )
4
6
A
( ? ,
? )
?
?
B
( ? ,
? )
4
6
A
( ? ,
? )
B
( ? ,
? )
G
?
?
?
?
Y
3
6
R
?
?
b)
G represents green
Y represents yellow
R represents red
A represents letter A
B represents letter B
Use the multiplication rule of probability to find the probability of winning
the game in one try.
P(winning the game) 5 P(Y, B)
5 P(Y) · P(B)
5
?
5
?
·
?
The probability of winning the game in one try is
? .
Lesson 11.3 Independent Events
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Technology Activity
Materials:
• spreadsheet software
SIMULATE RANDOMNESS
Work in pairs.
Background
Two fair six-sided number dice are thrown. Using a spreadsheet, you can generate
data to investigate how frequently the outcome of doubles (1 and 1, 2 and 2, … ,
6 and 6) occurs.
ST E P
1 Label your spreadsheet as shown.
ST E P
2 To generate a random integer between 1 and 6 in cell A2, enter the formula
5 INT(RAND()*611) to simulate rolling a die. A random number from 1 to 6
should appear in the cell.
ST E P
3 To model 100 rolls, select cells A2 to A101 and choose Fill Down from the Edit
menu.
ST E P
ST E P
ST E P
4 Repeat 2 and 3 for cells B2 to B101.
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ST E P
5 In cell C2, enter the formula 5 A2 2 B2. Select cells C2 to C101 and choose Fill
Down from the Edit menu. This column serves as a check to see if the random
numbers generated in columns A and B are the same. If the numbers are the
same, their difference is 0. A zero difference indicates doubles outcome.
ST E P
6 To see how many times the data show doubles occurring, enter the formula
5 COUNTIF(C2:C101,0) in cell D1.
ST E P
7 Find the experimental probability of the occurrence of two number dice
showing the same number by dividing the number you get in cell D1 by the
total, 100 rolls.
Find the theoretical probability of rolling doubles with 2
fair number dice. Compare this theoretical probability with the experimental
probability you obtained in the spreadsheet simulation. Are these two values the
same?
When you use a greater
number of simulations, such as
100 instead of 20, the result is
more likely to be closer to the
theoretical probability.
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Example 7
Solve probability problems involving independent events with
replacement.
A jar contains 8 green marbles and 4 red marbles. One marble is randomly drawn
and the color of the marble is noted. The marble is then put back into the jar and a
second marble is randomly drawn. The color of the second marble is also noted.
a) Find the probability of first drawing a green marble followed by a red marble.
Solution
1st Draw
8
12
2nd Draw
Outcome
G
(G, G)
Since the first marble is
drawn and replaced, the
probability of drawing
G
8
12
4
12
the second marble
4
12
R
(G, R)
8
12
G
(R, G)
remains unchanged.
R
4
12
R
G represents green
R represents red
(R, R)
P(G, R) 5 P(G) · P(R)
5
8 4
2
·
5
12 12
9
2
9
The probability of first drawing a green marble followed by a red marble is .
b) Find the probability of first drawing a red marble followed by a green marble.
Solution
P(R, G) 5 P(R) · P(G)
5
4 8
2
·
5
12 12
9
2
9
The probability of first drawing a red marble followed by a green marble is .
c)
Find the probability of drawing two green marbles.
Solution
P(G, G) 5 P(G) · P(G)
5
8 8
4
·
5
12 12
9
4
9
The probability of drawing two green marbles is .
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Guided Practice
Solve. Show your work.
2 In a bag, there are 9 magenta balls and 1 orange ball. Two balls are randomly
drawn, one at a time with replacement.
a) Find the probability of drawing two magenta balls.
1st Draw
2nd Draw
Outcome
O
(O, O)
?
?
M
(O, M)
?
?
O
(M, O)
?
?
O
1
10
9
10
M
?
?
M
O represents orange
M represents magenta
(M, M)
P(M, M) 5 P(M) · P(M)
5
?
5
?
·
?
The probability of drawing two magenta balls is
? .
b) Find the probability of drawing an orange ball followed by a magenta ball.
P(O, M) 5 P(O) · P(M)
5
?
5
?
·
?
The probability of drawing an orange ball followed by a magenta ball is
c)
Find the probability of drawing an orange ball both times.
? .
P(O, O) 5 P(O) · P(O)
5
?
5
?
·
?
The probability of drawing an orange ball both times is
? .
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Use the Addition Rule of Probability to Solve Problems with
Independent Events.
You have learned how to use the multiplication rule of probability to find the
probability of one favorable outcome in a compound event. Now you will learn to
use the addition rule of probability to find the probability of more than one favorable
outcome in a compound event.
A jar contains 8 green marbles and 4 red marbles. One marble is randomly drawn
and the color of the marble is noted. The marble is then put back into the jar and a
second marble is randomly drawn. The color of the second marble is also noted.
1st Draw
2nd Draw
Outcome
8
12
G
(G, G)
4
12
R
(G, R)
8
12
G
(R, G)
G
8
12
4
12
R
4
12
R
(R, R)
G represents green
R represents red
Suppose you want to find the probability of drawing two marbles of the same color.
There are two favorable outcomes, (G, G) and (R, R), and they are mutually exclusive.
P(R, R) 5
4 4
·
12 12
5
P(G, G) 5
5
1
9
8 8
·
12 12
4
9
To find the probability of (R, R) or (G, G), you can find the sum of their probabilities.
Using the addition rule of probability:
P(same color) 5 P(R, R) 1 P(G, G)
1
4
5 1
9
9
5
5
9
Math Note
Because (R, R) and (G, G) are
mutually exclusive events, you can
add the probabilities to find the
probability of (R, R) or (G, G).
In general, for two mutually exclusive events A and B, the addition rule of
probability states that:
P(A or B) 5 P(A) 1 P(B)
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Example 8
Solve probability problems with independent events involving more
than one favorable outcome.
Alex is taking two tests. The probability of him passing each test is 0.8.
a)
Find the probability that Alex passes both tests.
Solution
Math Note
To draw a tree diagram, first find the probability that
Alex fails the test.
Let P represent pass and F represent fail.
P(F) 5 1 2 P(P)
5 1 2 0.8
5 0.2
1st Test
2nd Test
Recall that for two complementary
events, the sum of their probabilities
is 1.
Outcome
(0.8)
P
(P, P )
(0.2)
F
(P, F)
(0.8)
P
(F, P )
P
(0.8)
(0.2)
F
(0.2)
F
(F, F)
P represents pass
F represents fail
P(P, P ) 5 P(P) · P(P)
5 0.8 · 0.8
5 0.64
The probability that Alex passes both tests is 0.64.
b)
Find the probability that he passes exactly one
of the tests.
Solution
Using the addition rule of probability:
P((P, F) or (F, P)) 5 P(P, F) 1 P(F, P )
5 P(P) · P(F ) 1 P(F ) · P(P)
5 0.8 · 0.2 1 0.2 · 0.8
5 0.32
The probability that he passes exactly one of the
tests is 0.32.
“Alex passes exactly
one of the tests”
means that he either
passes the first test
or the second test.
So, there are two
possible cases.
Think Math
What is the probability of passing at
least one test? Show your reasoning.
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Guided Practice
Solve. Show your work.
3 On weekends, Carli either jogs (J) or plays tennis (T) each day, but never both. The
probability of her playing tennis is 0.75.
a) Find the probability that Carli jogs on both days.
Because J and T are complementary,
P(J) 5 1 2 P(T)
51–
5
?
?
Saturday
Sunday
( ? )
Outcome
J
( ? ,
? )
T
( ? ,
? )
J
( ? ,
? )
J
(0.75)
( ? )
(0.75)
( ? )
T
(0.75)
( ? ,
T
? )
J represents jog
T represents tennis
P(J, J) 5 P(J) · P(J)
5 ?
5
·
?
?
The probability that Carli jogs on both days is
? .
b) Find the probability that Carli jogs on exactly one of the days.
Using the addition rule of probability:
P(J, T) or P(T, J) 5 P(J, T) 1 P(T, J)
5 P(J) · P(T) 1 P(T) · P(J )
5
?
5
?
248
·
?
1
?
·
?
The probability that Carli jogs on exactly one of the days is
? .
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Practice 11.3
Draw a tree diagram to represent each situation.
1 Tossing a fair coin followed by drawing a marble from a bag of 3 marbles:
1 yellow, 1 green, and 1 blue
2 Drawing two balls randomly with replacement from a bag with 1 green ball and
1 purple ball
3 Drawing a ball randomly from a bag containing 1 red ball and 1 blue ball, followed
by tossing a fair six-sided number die
4 Tossing a fair coin twice
5 Reading or playing on each day of a weekend
6 On time or tardy for school for two consecutive days
Solve. Show your work.
7 Mindy is playing a game that uses the spinner shown below and a fair coin. An
outcome of 3 on the spinner and heads on the coin wins the game.
a)
Draw a tree diagram to represent all possible outcomes and the corresponding
probabilities.
b)
Find the probability of winning the game in one try.
c)
Find the probability of losing the game in one try.
8 There are 2 blue balls and 4 yellow balls in a bag. A ball is randomly drawn from
the bag, and it is replaced before a second ball is randomly drawn.
a)
Draw a tree diagram to represent all possible outcomes.
b)
Find the probability that a yellow ball is drawn first, followed by another
yellow ball.
c)
Find the probability that a yellow ball is drawn after a blue ball is drawn first.
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9 Jasmine has 3 blue pens and 2 green pens in her pencil case. She randomly selects a
pen from her pencil case, and replaces it before she randomly selects again.
a) Draw a tree diagram to represent all possible outcomes and the corresponding
probabilities.
b) Find the probability that she selects 2 blue pens.
c)
d) Find the probability that she selects 2 pens of the same color.
Find the probability that she selects 2 green pens.
10 Henry has 4 fiction books, 6 nonfiction books, and 1 Spanish book on his
bookshelf. He randomly selects two books with replacement.
a) Draw a tree diagram to represent all possible outcomes and the corresponding
probabilities.
b) Find the probability that he selects a fiction book twice.
c)
d) Find the probability that he first selects a fiction book, and then a nonfiction book.
Find the probability that he first selects a nonfiction book, and then a
Spanish book.
11 Andy tosses a fair six-sided number die twice. What is the probability of tossing
an even number on the first toss and a prime number on the second toss?
12 The probability that Fiona wakes up before 8 A.M. when she does not set her alarm
2
5
is . On any two consecutive days that Fiona does not set her alarm, what is the
250
probability of her waking up before 8 A.M. for at least one of the days?
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13 A globe is spinning on a globe stand. The globe’s surface is painted 30% yellow,
10% green, and the rest is painted blue. Two times Danny randomly points to a spot
on the globe while it spins. The color he points to each time after the spinning stops
is recorded.
a) What is the probability that he points to the same color on both spins?
b) What is the probability that he points to yellow at least one time?
14 Sally thinks that for two independent events, because
the occurrence of one event will not have any impact on the probability of
the other event, they are also mutually exclusive. Do you agree with her? Explain
your reasoning using an example.
15 A game is designed so that a player wins when the game piece lands on letter A.
The game piece begins on letter G. A fair six-sided number die is tossed. If the
number tossed is odd, the game piece moves one step counterclockwise. If the
number tossed is even, the game piece moves one step clockwise.
a) What is the probability that a player will win after tossing the number die once?
b) What is the probability that a player will win after tossing the number die twice?
A
M
G
E
G
E
A
M
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