Probability of Independent and
Dependent Events and Review
Probability & Statistics
1.0 Students know the definition of the notion of independent
events and can use the rules for addition, multiplication, and
complementation to solve for probabilities of particular events in
finite sample spaces.
2.0 Students know the definition of conditional probability and
use it to solve for probabilities in finite sample spaces.
Probability of Independent and
Dependent Events and Review
Objectives
• Solve for the probability of
an independent event.
• Solve for the probability of a
dependent event.
Key Words
• Independent Events
– The occurrence of one event
does not affect the occurrence
of the other
• Dependents Events
– The occurrence of one event
does affect the occurrence of
the other
• Conditional Probability
– Two dependent events A and B,
the probability that B will occur
given that A has occurred.
Example 1
Identify Events
Tell whether the events are independent or dependent.
Explain.
a. Your teacher chooses students at random to
present their projects. She chooses you first, and
then chooses Kim from the remaining students.
b. You flip a coin, and it shows heads. You flip the coin
again, and it shows tails.
c. One out of 25 of a model of digital camera has some
random defect. You and a friend each buy one of the
cameras. You each receive a defective camera.
Example 1
Identify Events
SOLUTION
a. Dependent; after you are chosen, there is one fewer
student from which to make the second choice.
b. Independent; what happens on the first flip has no
effect on the second flip.
c. Independent; because the defects are random,
whether one of you receives a defective camera has
no effect on whether the other person does too.
Checkpoint
Identify Events
Tell whether the events are independent or dependent.
Explain.
1. You choose Alberto to be your lab partner. Then Tia
chooses Shelby.
ANSWER
dependent
2. You spin a spinner for a board game, and then you roll
a die.
ANSWER
independent
Example 2
Find Conditional Probabilities
Concerts A high school has a total of 850 students. The
table shows the numbers of students by grade at the
school who attended a concert.
a. What is the probability that a student at the school
attended the concert?
b. What is the probability that a junior did not attend the
concert?
Did not attend
Grade
Attended
Freshman
Sophomore
Junior
80
120
132
173
86
29
Senior
179
51
Example 2
Find Conditional Probabilities
SOLUTION
a. P(attended) =
total who attended
80 + 132 + 173 + 179
=
850
total students
564
~ 0.664
=
850
b. P(did not attend junior) =
juniors who did not attend
total juniors
29
29
~ 0.144
=
=
173 + 29
202
Checkpoint
Find Conditional Probabilities
3. Use the table below to find the probability that a
student is a junior given that the student did not
attend the concert.
ANSWER
Did not attend
Grade
Attended
Freshman
Sophomore
Junior
80
120
132
173
86
29
Senior
179
51
29
~ 0.101
286
Probability of
Independent and
Dependent Events
• Independent Events
– If A and B are independent
events, then the probability
that both A and B occur is
P(A and B)=P(A)*P(B)
• Dependent Events
– If A and B are dependent
events, then the probability
that both A and B occur is
P(A and B)=P(A)*P(B|A)
Example 3
Independent and Dependent Events
Games A word game has 100 tiles, 98 of which are letters
and two of which are blank. The numbers of tiles of each
letter are shown in the diagram. Suppose you draw two
tiles. Find the probability that both tiles are vowels in the
situation described.
a. You replace the first tile before
drawing the second tile.
b. You do not replace the first tile
before drawing the second tile.
Example 3
Independent and Dependent Events
SOLUTION
a. If you replace the first tile before selecting the second,
the events are independent. Let A represent the first
tile being a vowel and B represent the second tile
being a vowel. Of 100 tiles, 9 + 12 + 9 + 8 + 4 = 42
are vowels.
P( A and B = P( A • P( B =
42
42
•
= 0.1764
100 100
(
(
(
Example 3
Independent and Dependent Events
b. If you do not replace the first tile before selecting the
second, the events are dependent. After removing the
first vowel, 41 vowels remain out of 99 tiles.
P( A and B = P( A • P( B| A =
42
41 ~
0.1739
•
100 99
(
(
(
Checkpoint
Find Probabilities of Independent and
Dependent Events
4. In the game in Example 3, you draw two tiles. What is
the probability that you draw a Q, then draw a Z if you
first replace the Q? What is the probability that you
draw both of the blank tiles (without replacement)?
ANSWER
1
= 0.0001;
10,000
1 ~
0.0002
4950
Conclusion
Summary
• How are probabilities
calculated for two events
when the outcome of the
first event influences the
outcome of the second
event?
– Multiply the probability of the
second event, given that the
first event happen.
Assignment
• Probability of Independent
and Dependent Events
– Page 572
– #(11-14,15,18,22,26,30)
Review
Probability & Statistics
1.0 Students know the definition of the notion of independent events and
can use the rules for addition, multiplication, and complementation to solve
for probabilities of particular events in finite sample spaces.
2.0 Students know the definition of conditional probability and use it to
solve for probabilities in finite sample spaces.
Theoretical
Probability of an
Event
Example:
What is the probability
that the spinner shown
lands on red if it is equally
likely to land on any
section?
Solution:
The 8 sections represent the 8
possible outcomes. Three
outcomes correspond to the event
“lands on red.”
When all outcomes are equally
likely, the theoretical probability
that an event A will occur is:
Number of outcomes in event 𝐴
𝑃 𝐴 =
Total number of outcomes
The theoretical probability of an
event is often simply called its
probability.
3
Number of outcomes in event
P(red) =
=
8
Total number of outcomes
Experimental
Probability of an
Event
Example:
Surveys The graph shows results
of a survey asking students to
name their favorite type of footwear.
What is the experimental probability
that a randomly chosen student
prefers
(a) Sneakers?
(b) Shoes or boots?
For a given number of trials
of an experiment, the
experimental probability that
an event A will occur is:
Number of trials where 𝐴 𝑜𝑐𝑐𝑢𝑟𝑠
𝑃 𝐴 =
Total number trials
Solution:
Find the total number of students surveyed.
820+556+204+120=1700
a. Of 1700 students, 820 prefer sneakers.
P(prefers sneakers)
820
Number preferring sneakers
=
Total number of students
~
~ 0.48
=
1700
b. Of 1700 students surveyed, prefer shoes or boots.
P(prefers shoes or boots)
=
340
Number preferring shoes or boots
Total number of students
=
1700
~
~ 0.19
Probability of
Compound Events
• Overlapping Events
– If A and B are overlapping
events, then P(A and B)≠0,
and the probability of A or B
is:
𝑃 𝐴 𝑜𝑟 𝐵 = 𝑃 𝐴 + 𝑃 𝐵 − 𝑃(𝐴 𝑎𝑛𝑑 𝐵)
• Disjoint Events
– If A and B are disjoint
events, then P(A and B)=0,
and the probability of A or B
is:
𝑃 𝐴 𝑜𝑟 𝐵 = 𝑃 𝐴 + 𝑃 𝐵
Probability of the
Complement of an
Event
Recall:
Complement of an Event
All outcomes that are not
in the event
The sum of the probabilities
of an event and its
complement is 1.
𝑃 𝐴 + 𝑃 𝑛𝑜𝑡 𝐴 = 1
So,
𝑃 𝑛𝑜𝑡 𝐴 = 1 − 𝑃(𝐴)
Probability of
Independent and
Dependent Events
• Independent Events
– If A and B are independent
events, then the probability
that both A and B occur is
P(A and B)=P(A)*P(B)
• Dependent Events
– If A and B are dependent
events, then the probability
that both A and B occur is
P(A and B)=P(A)*P(B|A)