MATH 10: Elementary Statistics and Probability
Chapter 3: Probability Topics
Tony Pourmohamad
Department of Mathematics
De Anza College
Spring 2015
Probability
Basic Rules of Probability
Contingency Tables
Tree Diagrams
Objectives
By the end of this set of slides, you should be able to:
1
Understand and use the terminology of probability
2
Determine whether two events are mutually exclusive and whether
two events are independent
3
Calculate probabilities using Addition and Multiplication rules
4
Construct and interpret tree diagrams
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Probability
Basic Rules of Probability
Contingency Tables
Tree Diagrams
Some Necessary Terminology
• Experiment: A planned operation carried out under controlled
conditions
• Outcome: A result of an experiment
• Sample Space: Set of all possible outcomes of an experiment
• Event: Any combination of outcomes; A subset of the sample
space
• Example: Flipping a fair coin is an example of an experiment
. Outcome: You either get a heads (H) or a tails (T)
. Sample Space: S = {H , T }
. Event: You get a heads
• Other examples?
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Probability
Basic Rules of Probability
Contingency Tables
Tree Diagrams
So What Is Probability?
• This is like a debate on religion or philosophy in the field of statistics
• There is no one way to define probability
• So in this course how will we define probability?
Probability
. The probability of any outcome is the long-term relative frequency of
that outcome
. Probability is the likelihood or chance that something will happen
. A probability is a number between 0 and 1, inclusive
• Notation: P (A) is the probability that the event A occurs
• Example: A ="The event of getting a heads". So P (A) is the
probability of getting a heads.
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Probability
Basic Rules of Probability
Contingency Tables
Tree Diagrams
Equally Likely Events
• Equally likely means that each outcome of an experiment occurs
with equal probability
• Example: If you toss a fair six sided die, each face (1,2,3,4,5 or 6) is
as likely to occur as any other face
• Example: If you toss a fair coin, a Head (H) and a Tail (T) are
equally likely to occur
• To calculate the probability of an event A when all outcomes in
the sample space are equally likely, count the number of
outcomes for event A and divide by the total number of outcomes
in the sample space
P (A) =
number of outcomes for event A
total number of outcomes in S
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Probability
Basic Rules of Probability
Contingency Tables
Tree Diagrams
Equally Likely Events (continued)
• Example: Tossing a Fair Coin Twice
A fair coin has equal probability of landing on Head (H) or Tail (T).
Find the probability of getting ONE HEAD in two tosses:
. Sample space of outcomes for tossing a coin TWICE:
S = {HH , HT , TH , TT }
. A ="Getting ONE HEAD in two tosses"
. A = {HT , TH }
P ( A) =
=
number of outcomes for event A
total number of outcomes in S
2
4
= 0. 5
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Probability
Basic Rules of Probability
Contingency Tables
Tree Diagrams
Example #1
• Imagine rolling 1 fair die
• The sample space S = {1, 2, 3, 4, 5, 6}
Event
Event
Probability
Odd
A = {1, 3, 5}
P (A) = 63
Even
B = {2, 4, 6}
P (B ) = 36
2 or 4
D = { 2, 4}
P (D ) = 26
number≤ 4
T = { 1, 2, 3, 4}
P (T ) = 46
• Are all the events equally likely?
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Probability
Basic Rules of Probability
Contingency Tables
Tree Diagrams
Compound Events
• Compounds events are created using "AND" or "OR"
• "AND" event: "A and B" means BOTH events A and B occur
. The outcome that occurs satisfies both events A and B
. Event "A and B" includes items in common to both (intersection of)
A and B
. Example: Let A = {1, 2, 3, 4, 5} and B = {4, 5, 6, 7, 8}, then
"A and B" = {4, 5}
• "OR" event: "A or B" means either event A occurs or event B occurs
or both occur
. The outcome that occurs satisfies event A or event B or both
. Event "A or B" is the union of items from these events
. Example: Let A = {1, 2, 3, 4, 5} and B = {4, 5, 6, 7, 8}, then
"A or B" = {1, 2, 3, 4, 5, 6, 7, 8}
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Probability
Basic Rules of Probability
Contingency Tables
Tree Diagrams
Complements
• The complement of an event A is denoted A0 (read "A prime")
• A0 means that the event A did NOT occur
• A0 consists of all outcomes that are NOT in A
• Example: Let S = {1, 2, 3, 4, 5, 6} and let A = {1, 2, 3, 4}. Then
A0 = {5, 6}
• Example: If A is the event that it rains tomorrow, then A0 is the
event that it does NOT rain tomorrow
• Complement Rule: P (A) + P (A0 ) = 1 =⇒ P (A) = 1 − P (A0 )
• Example: Let S = {1, 2, 3, 4, 5, 6} and let A = {1, 2, 3, 4}. Then
A0 = {5, 6} and furthermore P (A) = 46 and P (A0 ) = 62 so
P (A) + P (A0 ) =
4
6
+
2
6
=1
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Probability
Basic Rules of Probability
Contingency Tables
Tree Diagrams
Conditional Probability
• Probability that event A occurs if we know that outcome B has
occurred
. We say this as "given that":
. We write this using a line | that means: given that
• P (A|B ) = "Probability that event A occurs given that outcome B
has occurred"
• P (event|condition)
. The outcome that we know has occurred is called the condition
. The condition is after the | line
. The condition reduces the sample space to be smaller by
eliminating outcomes that did not occur
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Probability
Basic Rules of Probability
Contingency Tables
Tree Diagrams
Example #2: Tossing a Fair Coin Twice
• A fair coin has equal probability of landing on Head (H) or Tail (T)
• Sample space of outcomes for tossing a coin TWICE:
S = {HH , HT , TH , TT }
1
2
Find the probability of getting TWO HEADS in two tosses of the coin:
So what is A?
1
P (A) =
4
Find the probability of getting TWO HEADS in two tosses IF WE
KNOW THAT (GIVEN THAT) the FIRST TOSS WAS A HEAD.
A = the event of getting two heads in two tosses
B = the first toss was a head
S = {HH , HT }, so
1
P (A|B ) =
2
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Probability
Basic Rules of Probability
Contingency Tables
Tree Diagrams
Example #2 Continued
3
Find the probability of getting ONE HEAD in two tosses.
A = the event of getting ONE HEAD in two tosses
P (A) =
4
2
4
=
2
Find the probability of getting ONE HEAD in two tosses GIVEN THAT
the FIRST TOSS WAS A HEAD.
A = the event of getting ONE HEAD in two tosses
B = the first toss was a head
S = {HH , HT }, so
P (A|B ) =
5
1
1
2
Find the probability of getting ONE HEAD in two tosses, GIVEN
THAT AT LEAST ONE HEAD was obtained.
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Probability
Basic Rules of Probability
Contingency Tables
Tree Diagrams
Example #2 Continued
5
Find the probability of getting ONE HEAD in two tosses, GIVEN
THAT AT LEAST ONE HEAD was obtained.
A = the event of getting ONE HEAD in two tosses
B = at least one head was obtained
S = {HH , HT , TH }, so
P (A|B ) =
2
3
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Probability
Basic Rules of Probability
Contingency Tables
Tree Diagrams
Example #3: Rolling a Fair Die
• Imagine rolling 1 fair die
• The sample space S = {1, 2, 3, 4, 5, 6}
Event
Event
Probability
1
Odd
A = {1, 3, 5}
P (A) = 63
Even
B = {2, 4, 6}
P (B ) = 36
2 or 4
D = { 2, 4}
P (D ) = 26
number≤ 4
T = { 1, 2, 3, 4}
P (T ) = 46
Find the probability of rolling a number ≤ 4 GIVEN THAT the
outcome is even
P (T |B ) = P (1 or 2 or 3 or 4|2 or 4 or 6) =
2
3
. The "reduced" (smaller) sample space is S = {2, 4, 6}; odd
numbers were removed from the sample space
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Probability
Basic Rules of Probability
Contingency Tables
Tree Diagrams
Example #3 Continued: Rolling a Fair Die
2
Use the reduced sample space to find the probability of rolling an
odd number given that the outcome is ≤ 4.
P (odd| ≤ 4) = P (A|T ) = P (1 or 3 or 5|1 or 2 or 3 or 4) =
3
2
4
=
1
2
Find the probability of rolling an odd number
P (odd) = P (A) =
3
6
=
1
2
• Is the event in 2) more likely to occur than the event in 3)?
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Probability
Basic Rules of Probability
Contingency Tables
Tree Diagrams
Important to Note
• For AND and OR events, the order of listing the events does not
matter and can be switched, i.e.,
P (A and B ) = P (B and A)
and
P (A or B ) = P (B or A)
• For CONDITIONAL PROBABILITY the order is important, i.e.,
P (A|B ) 6= P (B |A)
in most situations
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Probability
Basic Rules of Probability
Contingency Tables
Tree Diagrams
Mutually Exclusive Events
• Mutually Exclusive Events: Two events A and B that cannot occur
at the same time are said to be mutually exclusive (also called
disjoint)
• If two events are mutually exclusive, then P (A and B ) = 0
• Examples:
. List a pair of events above that are mutually exclusive and are not
complements
. List a pair of events above that are mutually exclusive and are
complements
. Are being a part-time and a full-time student at De Anza mutually
exclusive?
. Are being a day student and a night student at De Anza mutually
exclusive?
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Probability
Basic Rules of Probability
Contingency Tables
Tree Diagrams
Independent Events
• Independent Events: Two events, A and B, are independent if and
only if the probability of one event occurring is not affected by
whether the other event occurs or not
• If A and B are independent, then P (A|B ) = P (A) and
P (B |A) = P (B ). Why?
• If two events are NOT independent, then we say that they are
dependent
• Example: Two events that are independent
. A = the event of getting a HEAD on a coin flip
. B = the event of getting a 4 when rolling a die
• Other examples of independent or dependent events?
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Probability
Basic Rules of Probability
Contingency Tables
Tree Diagrams
With and Without Replacement
• With Replacement: If each member of the population is replaced
after it is picked, then that member has the possibility of being
chosen more than once.
• When sampling is done with replacement, the events are
independent
• Without Replacement: When sampling is done without
replacement, each member of a population may be chosen only
once
• When sampling is done without replacement, the events are
dependent
• Examples?
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Probability
Basic Rules of Probability
Contingency Tables
Tree Diagrams
With and Without Replacement
• Example: I have a bag with 4 blue marbles and 6 red marbles.
What is the probability of pulling out a blue marble?
A = pulling out a blue marble
S = {B1 , B2 , B3 , B4 , R1 , R2 , R3 , R4 , R5 , R6 }
P (A) =
4
10
• Now imagine I pull marbles out of the bag without replacement,
and on the first draw I pulled out a blue marble, what is the
probability of pulling out a blue marble now?
P (A) =
3
9
• Why? What would have happened if I had done it with
replacement?
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Probability
Basic Rules of Probability
Contingency Tables
Tree Diagrams
Mutually Exclusive Versus Independence
• Note that the concepts of mutually exclusive and independent
are not equivalent
• For example, two roles of a fair die are independent events,
however, they are not mutually exclusive
• Two events that are not mutually exclusive may or may not be
independent
• Two events that are mutually exclusive must be dependent
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Probability
Basic Rules of Probability
Contingency Tables
Tree Diagrams
Example #4
• Are the events independent? Show justification using probabilities
and state your conclusion
Event
Event
Probability
1
Odd
A = {1, 3, 5}
P (A) = 63
Even
B = {2, 4, 6}
P (B ) = 36
2 or 4
D = { 2, 4}
P (D ) = 26
number≤ 4
T = { 1, 2, 3, 4}
P (T ) = 46
Are events A = "odd number" and T = "number ≤ 4"
independent?
2
1
P (A|T ) = =
4
2
and
1
3
P (A) = =
6
2
Conclusion: The events A and T are independent since P (A|T ) = P (A)
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Probability
Basic Rules of Probability
Contingency Tables
Tree Diagrams
Example #5
Which of the following describe independent events?
• Repeated tosses of a coin
• Selecting 2 cards consecutively from a deck of 52 cards, without
replacement
• Selecting 2 cards from a deck of cards, with replacement
• The numbers that show on each of two dice when tossed
• The color of two marbles selected consecutively from a jar of
colored marbles, without replacement
• The color of two marbles selected from a jar of colored marbles,
with replacement
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Probability
Basic Rules of Probability
Contingency Tables
Tree Diagrams
Example #6
• In India, adult (15+ years) literacy rates are 82.1% for men and
65.5% for women The overall literacy rate is estimated as
approximately 74%.
• Is the literacy rate in India independent of gender? Justify your
answer using appropriate probabilities.
• Consider the population of residents of India age 15 and over:
Events: F = female, M = male, L = literate
Now,
P (L|M ) = 82.1%
yet
P (L) = 74%
so P (L|M ) 6= P (L) and so they are not independent.
• Source: http://www.censusindia.gov.in/2011-prov-results/indiaatglance.html
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Probability
Basic Rules of Probability
Contingency Tables
Tree Diagrams
Two Basic Rules of Probability
• When calculating probability, there are two rules to consider when
determining if two events are independent or dependent and if
they are mutually exclusive or not
• The rules
The Addition Rule
and
2 The Multiplication Rule
1
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Probability
Basic Rules of Probability
Contingency Tables
Tree Diagrams
The Addition Rule
• The Addition Rule: Let A and B be events, then
P (A or B ) = P (A) + P (B ) − P (A and B )
• If A and B are mutually exclusive events, then
P (A or B ) = P (A) + P (B )
• Example: Imagine flipping a fair coin twice
. A = the event of getting a HEADS (H) on the first flip
. B = the event of getting a HEADS (H) on the second flip
. A and B are not mutually exclusive, why?
. So what is the P (A or B)? Does it make sense?
P (A or B ) = P (A) + P (B ) − P (A and B ) =
1
2
+
1
2
−
1
4
=
3
4
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Probability
Basic Rules of Probability
Contingency Tables
Tree Diagrams
Example #7: Rolling Two Dice
• The sample space for when rolling two dice is
(1,1) (1,2) (1,3) (1,4) (1,5)
(2,1) (2,2) (2,3) (2,4) (2,5)
(3,1) (3,2) (3,3) (3,4) (3,5)
(4,1) (4,2) (4,3) (4,4) (4,5)
(5,1) (5,2) (5,3) (5,4) (5,5)
(6,1) (6,2) (6,3) (6,4) (6,5)
(1,6)
(2,6)
(3,6)
(4,6)
(5,6)
(6,6)
• What is the probability of a rolling a sum of 6?
A = rolling a sum of 6
5
P (A) =
36
• What is the probability of rolling a double (i.e., the same numbers)?
B = rolling a double
6
P (B ) =
36
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Probability
Basic Rules of Probability
Contingency Tables
Tree Diagrams
Example #7 Continued: Rolling Two Dice
(1,1)
(2,1)
(3,1)
(4,1)
(5,1)
(6,1)
(1,2)
(2,2)
(3,2)
(4,2)
(5,2)
(6,2)
(1,3)
(2,3)
(3,3)
(4,3)
(5,3)
(6,3)
(1,4)
(2,4)
(3,4)
(4,4)
(5,4)
(6,4)
(1,5)
(2,5)
(3,5)
(4,5)
(5,5)
(6,5)
(1,6)
(2,6)
(3,6)
(4,6)
(5,6)
(6,6)
• What is the probability of a rolling a sum of 6 or a double?
A = rolling a sum of 6
B = rolling a double
P (A or B ) = P (A) + P (B ) − P (A and B )
=
5
36
+
6
36
−
1
36
=
10
36
• Are A and B mutually exclusive?
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Probability
Basic Rules of Probability
Contingency Tables
Tree Diagrams
The Multiplication Rule
• The Multiplication Rule: Let A and B be events, then
P (A and B ) = P (A|B )P (B )
• If A and B are independent events, then
P (A and B ) = P (A)P (B )
• Example: Imagine flipping a fair coin twice
. A = the event of getting a HEADS (H) on the first flip
. B = the event of getting a HEADS (H) on the second flip
. A and B are independent, why?
. So what is the P (A and B)? Does it make sense?
P (A and B ) = P (A)P (B ) =
1
2
×
1
2
=
1
4
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Probability
Basic Rules of Probability
Contingency Tables
Tree Diagrams
Example #8: Bag of Marbles
• Imagine you are picking marbles out of a bag without
replacement
• There are 4 blue marbles and 6 red marbles
• What is the probability of selecting a blue marble on the 1st draw
and a red marble on the 2nd?
B = selecting a blue marble
R = selecting a red marble
• Are these events independent?
P (B and R ) = P (B )P (R |B ) =
4
10
×
6
9
=
24
90
• What about the same problem but with replacement?
P (B and R ) = P (B )P (R ) =
4
10
×
6
10
=
24
100
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Probability
Basic Rules of Probability
Contingency Tables
Tree Diagrams
Example #8 Continued: Bag of Marbles
• Imagine picking marbles out of a bag without replacement
• There are 4 blue marbles and 6 red marbles
• What is the probability of selecting a blue marble and a red
marble (in any order)?
B = selecting a blue marble
R = selecting a red marble
P (B and R ) = P (B on 1st and R on 2nd OR R on 1st and B on 2nd)
= P (B on 1st and R on 2nd) + P (R on 1st and B on 2nd)
=
4
10
×
6
9
+
6
10
×
4
9
=
48
90
• Why didn’t I subtract
P (B on 1st and R on 2nd AND R on 1st and B on 2nd)
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Probability
Basic Rules of Probability
Contingency Tables
Tree Diagrams
The Conditional Probability Rule
• The Conditional Probability Rule: If P (B ) 6= 0 then
P (A|B ) =
P (A and B )
P (B )
• Why is this true?
• Recall the multiplication rule: P (A and B ) = P (A|B )P (B )
If I multiply both sides by 1/P (B ) then I obtain
1
P (B )
× P (A and B) = P (A|B)P (B) ×
1
P (B )
P (A|B )
P (A and B )
P (
B
)
=
=⇒
P (B )
P (B )
P (A and B )
=⇒
= P (A|B)
P (B )
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Probability
Basic Rules of Probability
Contingency Tables
Tree Diagrams
Example #9
• In a certain neighborhood:
. 65% of residents subscribe to the Mercury News
. 30% of residents subscribe to the SF Chronicle
. 20% of residents subscribe to both newspapers
• Let M = person subscribes to the Mercury News and
C = person subscribes to the Chronicle
• Find the probability that a person subscribes to the Mercury News
given that he/she subscribes to the SF Chronicle
• What does this mean in the language of probability?
. We want to find P (M |C ), so using the conditional probability rule
P (M |C ) =
P (M and C )
p(C )
=
.20
2
=
.30
3
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Probability
Basic Rules of Probability
Contingency Tables
Tree Diagrams
Example #10: Bag of Marbles
• Imagine you are picking marbles out of a bag without
replacement
• There are 4 blue marbles and 6 red marbles
• What is the probability of selecting a blue marble on the 2nd draw
given that a red marble was selected on the 1st draw?
B = selecting a blue marble
R = selecting a red marble
P (B |R ) =
=
=
P (B and R )
P (R )
24
90
6
10
24
90
×
10
6
=
4
9
• Could I have solved this faster?
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Probability
Basic Rules of Probability
Contingency Tables
Tree Diagrams
Contingency Tables
• A contingency table displays data for two variables
• The contingency table shows the number of individuals or items in
each category
• Example of a contingency table:
Smoker
Non-Smoker
Total
Lunger Cancer
70
5
75
No Lung Cancer
20
5
25
Total
90
10
100
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Probability
Basic Rules of Probability
Contingency Tables
Tree Diagrams
Contingency Tables
• We can use the data in the table to find probabilities
• All probabilities, EXCEPT conditional probabilities, have the grand
total in the denominator
Smoker
Non-Smoker
Total
Lunger Cancer
70
5
75
No Lung Cancer
20
5
25
Total
90
10
100
• Example: What is the probability of not having lung cancer?
P (No Lung Cancer) =
25
100
=
1
4
• Example: What is the probability of being a smoker?
P (Being a smoker) =
90
100
=
9
10
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Probability
Basic Rules of Probability
Contingency Tables
Tree Diagrams
Contingency Tables
• The condition limits you to a particular row or column in the table
• The denominator will be the total for the row or column in the
table that corresponds to the condition
Smoker
Non-Smoker
Total
Lunger Cancer
70
5
75
No Lung Cancer
20
5
25
Total
90
10
100
• Example: What is the probability of not having lung cancer given
you were a smoker?
P (No Lung Cancer|Smoker) =
20
90
=
2
9
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Probability
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Tree Diagrams
Contingency Tables
Smoker
Non-Smoker
Total
Lunger Cancer
70
5
75
No Lung Cancer
20
5
25
Total
90
10
100
• Example: What is the probability of not having lung cancer and
being a smoker?
P (No Lung Cancer and Smoker) =
20
100
=
2
10
alternatively,
P (No Lung Cancer and Smoker) = P (No Lung Cancer|Smoker)P (Smoker )
=
=
2
9
2
×
9
10
10
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Probability
Basic Rules of Probability
Contingency Tables
Tree Diagrams
Contingency Tables
Smoker
Non-Smoker
Total
Lunger Cancer
70
5
75
No Lung Cancer
20
5
25
Total
90
10
100
• Example: What is the probability of not having lung cancer or
being a smoker?
P (No Lung Cancer or Smoker) = P (No Lung Cancer) + P (Smoker) − P (No Lung Cancer and Smoker)
25
90
20
=
=
100
95
+
100
−
100
100
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Probability
Basic Rules of Probability
Contingency Tables
Tree Diagrams
Example #11
A large car dealership examined a sample of vehicles sold or leased in
the past year.
Car (C) SUV (S) Van (V) Truck (T) Total
New Vehicle (N)
86
25
21
38
170
Used Vehicle (U)
49
13
4
22
78
Leased Vehicle (L)
34
12
6
0
52
Total
159
50
31
60
300
• Example: What is the probability that the vehicle was leased?
P (L) =
52
300
• Example: What is the probability that the vehicle was a SUV?
P (S ) =
50
300
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Probability
Basic Rules of Probability
Contingency Tables
Tree Diagrams
Example #11 Continued
A large car dealership examined a sample of vehicles sold or leased in
the past year.
Car (C) SUV (S) Van (V) Truck (T) Total
New Vehicle (N)
86
25
21
38
170
Used Vehicle (U)
49
13
4
22
78
Leased Vehicle (L)
34
12
6
0
52
Total
159
50
31
60
300
• Example: What is the probability that the vehicle was a truck?
P (T ) =
60
300
• Example: What is the probability that the vehicle was not a truck?
P (T 0 ) = 1 − P (T ) = 1 −
60
300
=
240
300
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Probability
Basic Rules of Probability
Contingency Tables
Tree Diagrams
Example #11 Continued
• Test yourself:
Find the probability that the vehicle was a car AND was leased
Find the probability that the vehicle was used GIVEN THAT it was a
van
3 Find the probability that the vehicle was used OR was a van
1
2
• Solutions:
1 34/300
2 4/31
3 105/300 Hint: Use the addition rule
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Probability
Basic Rules of Probability
Contingency Tables
Tree Diagrams
Example #11 Continued
New Vehicle (N)
Used Vehicle (U)
Leased Vehicle (L)
Total
Car (C)
86
49
34
159
SUV (S)
25
13
12
50
Van (V)
21
4
6
31
Truck (T)
38
22
0
60
Total
170
78
52
300
• Example: Are the events N and V independent?
P (N |V ) =
and
P (N ) =
21
31
31
300
• P (N |V ) 6= P (N ) so not independent
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Probability
Basic Rules of Probability
Contingency Tables
Tree Diagrams
Tree Diagrams
• A Tree Diagram is a special type of graph used to determine the
outcomes of an experiment
• It consists of "branches" that are labeled with either frequencies or
probabilities
• Each "branch" represents a mutually exclusive outcome
• Tree diagrams can make some probability problems easier to
visualize and solve
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Probability
Basic Rules of Probability
Contingency Tables
Tree Diagrams
Example #12
• An urn contains 11 marbles, 3 Red and 8 Blue
• We are selecting 2 marbles randomly from the urn with
replacement
• Draw the tree diagram using frequencies.
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Probability
Basic Rules of Probability
Contingency Tables
Tree Diagrams
Example #12 Continued
• Why 9RR?
• Imagine we label all the marbles
{B1 , B2 , B3 , B4 , B5 , B6 , B7 , B8 , R1 , R2 , R3 }
• There are 9 unique ways I can draw two red marbles with
replacement, i.e.,
{(R1 , R1 )(R1 , R2 ), (R2 , R1 ), (R1 , R3 ), (R3 , R1 ), (R2 , R2 ),
(R2 , R3 ), (R3 , R2 ), (R3 , R3 )}
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Probability
Basic Rules of Probability
Contingency Tables
Tree Diagrams
Example #12 Continued
• Could also write the tree using probabilities
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Probability
Basic Rules of Probability
Contingency Tables
Tree Diagrams
Example #12 Continued
• What do each of those values in the previous slide mean?
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Probability
Basic Rules of Probability
Contingency Tables
Tree Diagrams
Example #12 Continued
• What is the probability of getting a blue marble on the 1st draw
and a blue marble on the 2nd?
• What is the probability of getting a red marble on the 1st draw
and a blue marble on the 2nd?
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Probability
Basic Rules of Probability
Contingency Tables
Tree Diagrams
Example #12 Continued
• What is the probability of getting a blue marble on the 1st draw
OR a red marble on the 1st draw?
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Probability
Basic Rules of Probability
Contingency Tables
Tree Diagrams
Example #12 Continued
• What is the probability of getting a red on the 1st and blue on the
2nd OR a blue on the 1st or red on the 2nd?
P (RB or BR ) =
3
11
×
8
10
+
8
11
×
3
10
=
48
10
51 / 53
Probability
Basic Rules of Probability
Contingency Tables
Tree Diagrams
Example #12 Continued
• What is the probability of getting a red on the 2nd given that you
picked a blue on the 1st?
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Probability
Basic Rules of Probability
Contingency Tables
Tree Diagrams
Tree Diagrams: Some Things to Remember
• Each complete path through the tree represents a separate
mutually exclusive outcome in the sample space
• Some steps
Draw a tree representing the possible mutually exclusive outcomes
Assign conditional probabilities along the branches of the tree
3 Multiply probabilities along each complete path through the tree
to find probabilities of each "AND" outcome in the sample space
4 Add probabilities for the appropriate paths of a tree to find the
probability of a compound OR event
1
2
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