11.4
Dependent Events
Lesson Objectives
Vocabulary
• Understand dependent events.
• Use the rules of probability to solve problems with dependent events.
dependent events
Understand Dependent Events.
Suppose there are 3 yellow cards and 2 red cards. They are shuffled and placed in
a stack. You are asked to draw two cards randomly, one at a time, from the stack
without looking at the cards.
When the first card is drawn, it is not replaced. So when the second card
is drawn, there are only 4 cards left in the stack. The probability of
drawing a particular color card will change after the first draw because
the sample space for the second event changes. In the first draw, there
are 5 cards while in the second draw, there are 4 cards left.
When the occurrence of one event causes the probability of another
event to change, the two events are said to be dependent.
Y
Y
R
R
Y
You can represent the dependent events described above with a tree diagram, as
shown.
1st Draw
2nd Draw
2
4
Y
Outcome
(Y, Y)
Notice how the probabilities
of the outcomes for the
second event change for
dependent events.
Y
3
5
2
4
2
5
3
4
R
(Y, R)
Y
(R, Y)
R
1
4
252
R
(R, R)
Y represents yellow
R represents red
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Use the Multiplication Rule of Probability to Solve Problems with
Dependent Events.
Consider the 5-card scenario again. To find the probability of drawing 2 red cards
one after another without replacement, first you locate the branches that will give the
favorable outcome (R, R). Then you multiply the probabilities along the branches. In
other words, you multiply the probability of drawing a red card in the first draw with
the probability of drawing a red card in the second draw.
1st Draw
2nd Draw
2
4
Outcome
Y
(Y, Y)
R
(Y, R)
Y
(R, Y)
Y
3
5
2
4
2
5
3
4
R
1
4
R
(R, R)
Y represents yellow
R represents red
P(R, R) 5 P(R) · P(R after R)
5
2 1
·
5 4
5
1
10
Math Note
For dependent events, you cannot
simply count the outcomes to find
the probability of the compound
events.
In general, for two dependent events A and B, the multiplication rule of
probability states:
P(A and B) 5 P(A) · P(B after A)
Lesson 11.4 Dependent Events
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Example 9
Understand dependent events.
Inside a jar, there are 3 blue marbles and 7 green marbles. Rena randomly draws two
marbles, one after another without replacement. Draw a tree diagram to represent
the possible outcomes of this compound event.
Solution
Let B represent blue and G represent green.
1st draw P(B) 5
3
10
P(G) 5
7
10
Math Note
Because the events are dependent,
the number of possible outcomes
for the second event is reduced by 1
after the first event occurs.
2nd draw 2
9
There are 2 blue marbles left after 1 blue marble is drawn.
7
9
There are 7 green marbles left after 1 blue marble is drawn.
3
9
There are 3 blue marbles left after 1 green marble is drawn.
6
9
There are 6 green marbles left after 1 green marble is drawn.
P(B after B) 5 P(G after B) 5 P(B after G) 5 P(G after G) 5 1st Draw
2nd Draw
2
9
Outcome
B
(B, B)
G
(B, G)
B
(G, B)
B
3
10
7
9
7
10
3
9
G
6
9
254
G
(G, G)
B represents blue
G represents green
Chapter 11 Probability
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Guided Practice
Solve. Show your work.
1 A deck of four cards with the letters D, E, E, D are placed facing down on a table.
Two cards are turned over at random to show the letters. Draw a tree diagram
to represent the possible outcomes for this compound event.
D
E
E
D
Let D represent the letter D and E represent the letter E.
1st draw P(D) 5
2
4
P(E) 5
2
4
2nd draw ?
?
There is
?
?
There are
?
E's left after 1 D is drawn.
?
?
There are
?
D’s left after 1 E is drawn.
?
?
There is
P(D after D) 5 P(E after D) 5 P(D after E) 5 P(E after E) 5 1st Draw
?
?
?
?
D left after 1 D is drawn.
E left after 1 E is drawn.
2nd Draw
Outcome
D
(D, D)
E
(D, E)
D
(E, D)
E
(E, E)
D
2
4
?
?
2
4
?
?
E
?
?
Lesson 11.4 Dependent Events
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Example 10 Solve a probability problem involving dependent events
without replacement.
A jar contains 8 green marbles and 4 red marbles. Two marbles are randomly drawn,
one at a time without replacement.
a) Find the probability of drawing a green marble followed by a red marble.
Solution
1st Draw
2nd Draw
Outcome
G
(G, G)
R
(G, R)
G
(R, G)
7
11
G
8
12
4
11
4
12
8
11
R
3
11
R
(R, R)
G represents green
R represents red
P(G, R) 5 P(G) · P(R after G)
5
8 4
·
12 11
5
8
33
The probability of randomly drawing a green marble followed by a red marble is
8
.
33
Recall that for two dependent events A
and B, P(A and B) 5 P(A) · P(B after A).
256
Chapter 11 Probability
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b)
Find the probability of randomly drawing a red marble followed by a green
marble.
Solution
P(R, G) 5 P(R) · P(G after R)
5
4 8
·
12 11
5
8
33
The probability of randomly drawing a red marble followed by a green marble is
c)
8
.
33
Find the probability of randomly drawing 2 green marbles.
Solution
P(G, G) 5 P(G) · P(G after G)
5
8 7
·
12 11
5
14
33
The probability of randomly drawing 2 green marbles is
d)
14
.
33
Find the probability of randomly drawing 2 red marbles.
Solution
P(R, R) 5 P(R) · P(R after R)
5
4 3
·
12 11
5
1
11
The probability of randomly drawing 2 red marbles is
1
.
11
Think Math
If there is just 1 red marble in the jar,
what is the probability of drawing
two red marbles? Justify your
answer.
Lesson 11.4 Dependent Events
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Guided Practice
Solve. Show your work.
2 There are 16 colored pebbles in a jar. 11 of them are blue and the rest are
orange. Two pebbles are randomly selected from the jar, one at a time without
replacement.
a) Find the probability of taking an orange pebble followed by a blue pebble.
1st Draw
2nd Draw
10
15
Outcome
B
(B, B)
O
(B, O)
B
(O, B)
B
11
16
5
15
5
16
11
15
O
4
15
O
(O, O)
B represents blue
O represents orange
P(O, B) 5 P(O) · P(B after O)
5
?
?
·
?
?
5
?
?
The probability of randomly taking an orange pebble followed by a blue pebble is
? .
b) Find the probability of taking two orange pebbles.
P(O, O) 5 P(O) · P(O after O)
5
?
?
·
?
?
5
?
?
The probability of randomly taking two orange pebbles is
c)
? .
Find the probability of taking two blue pebbles.
P(B, B) 5 P(B) · P(B after B)
258
5
?
?
·
?
?
5
?
?
The probability of randomly taking two blue pebbles is
? .
Chapter 11 Probability
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Example 11
Solve probability problems involving dependent events
with more than one favorable outcome.
Scott randomly chooses to go to school by either bus or bicycle, but not both. The
tree diagram below shows that Scott’s choice of transportation depends on the
weather. The probability that it rains on a particular day is denoted by a. Assume that
rainy and sunny days are mutually exclusive events.
Weather
Transportation
1
4
Remember that the
Bi
probabilities of all
R
(a)
(1 2 a)
branches from a node
3
4
Bu
2
3
Bi
1
3
Bu
must have a sum of 1.
R represents rainy day
S represents sunny day
Bi represents bicycle
Bu represents bus
S
1
a)If the probability that it rains is , find the probability that Scott will take a bus
2
to school on any day.
You need to find the probability
that Scott takes the bus on a
Solution
P(R) 5
rainy day plus the probability that
he takes the bus on a sunny day.
1
2
P(S) 5 1 2
P(Bu) 5
1
1
5 2
2
Events R and S are complementary.
1 3
1 1
13
· 1 · 5 2 4
2 3
24
Evaluate P(R, Bu) 1 P(S, Bu).
1
2
If the probability that it rains is , then the probability that Scott will take a bus to school is
13
.
24
5
8
b)If the probability that it rains is , find the probability that Scott will ride a bicycle to school.
Solution
P(S) 5 1 2
P(Bi) 5
5
3
5 8
8
5 1
3 2
13
· 1 · 5 8 4
8 3
32
Events R and S are complementary.
Evaluate P(R, Bi) 1 P(S, Bi).
5
8
If the probability that it rains is , then the probability that Scott will ride a bicycle is
13
.
32
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Guided Practice
Solve. Show your work.
3 The tree diagram below shows how passing a test depends on whether a student
studies (S) or does not study (NS) for the test. The probability that a student
studies is denoted by p. Assume that S and NS are mutually exclusive events.
Preparation
Result
2
3
P
1
3
F
1
5
P
4
5
F
S
(p)
(1 2 p)
NS
S represents study
NS represents does not study
P represents pass
F represents fail
a) If the probability of studying is 0.4, find the probability that a student passes the test.
P(S) 5
?
?
Write the fraction for 0.4.
P(NS) 5 1 2 P(S)Events S and NS are complementary.
512
5
P(P) 5
5
?
?
?
?
?
?
?
?
·
1
· ?
?
?
?
Evaluate P(S, P) 1 P(NS, P).
?
?
If the probability of studying is 0.4, then the probability that a student passes
the test is ? .
b) If the probability of studying is 0.75, find the probability that a student fails the test.
P(S) 5
?
?
Write the fraction for 0.75.
P(NS) 5 1 2 P(S)Events S and NS are complementary.
512
5
P(F) 5
5
260
?
?
?
?
?
?
?
?
·
1
· ?
?
?
?
Evaluate P(S, F) 1 P(NS, F).
?
?
If the probability of studying is 0.75, then the probability that a student fails
the test is ? .
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Practice 11.4
State whether each pair of events is dependent or independent.
1 Drawing 2 red balls randomly, one at a time without replacement, from a bag of
6 balls
2 Tossing a coin twice
3 Reaching school late or on time for two consecutive days
4 Flooding of roads during rainy or sunny days
Draw a tree diagram for each situation.
5 2 balls are drawn at random, one at a time without replacement, from a bag of
3 green balls and 18 red balls.
6 The probability of rain on a particular day is 0.3. If it rains, then the probability that
Renee goes shopping is 0.75. If it does not rain, then the probability that she goes
jogging is 0.72. Assume that shopping and jogging are mutually exclusive and
complementary, and that rain and no rain are complementary.
Solve. Show your work.
7 Geraldine has a box of 13 colored pens: 3 blue, 4 red, and the rest black. What is the
probability of drawing two blue pens randomly, one at a time without replacement?
8 A box contains 8 dimes, 15 quarters, and 27 nickels. A student randomly draws
two items, one at a time without replacement, from the bag. Find the probability that
2 quarters are drawn.
9 There are 9 green, 2 yellow, and 5 blue cards in a deck. Players A and B each
randomly draw a card from the deck. Player A draws a card first before Player B.
Find the probability that both players draw the same color cards.
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10 The probability diagram below shows the probability of Xavier going to a library
or a park depending if the weather is sunny or rainy. The probability of rain on a
particular day is denoted by a. Assume that going to the library and going to the
park are mutually exclusive and complementary.
Weather
Place
2
3
L
1
3
P
1
5
L
R
(a)
(1 2 a)
S
( ? )
P
R represents rainy
S represents sunny
L represents library
P represents park
a)If a 5 0.3, find the probability that Xavier goes to the park on any day.
b)If a 5 0.75, find the probability that he goes to the library on any day.
11 There are 15 apples in a fruit basket. 6 of them are red apples and
the rest are green apples. Two apples are randomly picked one at a
time without replacement.
a) Draw a tree diagram to represent all possible outcomes and the
corresponding probabilities.
b) Find the probability of picking a green apple and then a red apple.
c)
d) Find the probability of picking two red apples.
Find the probability of picking two green apples.
12 There are 8 people in a room: 3 of them have red hair, 2 have blonde hair, and the
rest have dark hair. Two people are randomly selected to leave the room, one after
another, and they do not re-enter the room.
a) Draw a tree diagram to represent all possible outcomes and the
corresponding probabilities.
b) What is the probability of a person with dark hair leaving the room first?
c)
d) What is the probability of two people with the same hair color leaving the room?
262
What is the probability of a person with red hair leaving the room, followed
by a person with blonde hair?
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13 Along a stretch of road, two intersections have traffic lights. Having a red or a green light
for the first intersection is equally likely. Having a red light at the second intersection is
twice as likely as a green light, if the traffic light is red at the first intersection. What is the
probability of having a red light at the first intersection and a green light at the second
intersection? Draw a tree diagram to show the possible outcomes.
14 To get to work, Mr. Killiney needs to take a train and then a bus. The probability that
the train breaks down is 0.1. When the train breaks down, the probability that the
bus will be overcrowded is 0.7. When the train is operating normally, the probability
that the bus will be overcrowded is 0.2. What is the probability that the bus will not
be overcrowded? Draw a tree diagram to show all possible outcomes.
1 If there are 12 green and 6 red apples, find the probability of randomly choosing
three apples of the same color in a row, without replacement. Show your work.
2 William has five $1 bills, ten $10 bills, and three $20 bills in his wallet. He picks
three bills randomly in a row, without replacement. What is the probability of
him picking three of the same type of bills? Show your work.
3 Daniel plans to visit Australia. Whether he goes alone or with a companion is
equally likely. If he travels with a companion there is a 40% chance of joining a
guided tour. If he travels alone, there is an 80% chance of joining a guided tour.
a)
What is the probability of Daniel traveling with a companion and not joining
a guided tour?
b)
What is the probability of Daniel joining a guided tour?
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