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Section 7.1 - Experiments, Sample Spaces, and Events
Definition: An experiment is an activity with an observable result.
Definition: The outcome is the result of an experiment.
Definition: The sample space is the set of all outcomes of an experiment.
Definition: An event is a subset of the sample space. (Note: An event E is said to occur whenever E contains
the observed outcome.)
/
Definition: E and F are mutually exclusive if E ∩ F = 0.
Note: All of the set operations (union, intersection, complement) work the same with events.
Example 1: Let’s consider the experiment of rolling a fair six-sided die and observing the number that lands
uppermost.
a) Determine the sample space.
b) Find the event E where E = {x|x is an even number}.
c) Find the event F where F = {x|x is a number greater than 2}
d) Find the event (E ∩ F)
e) Find the event (E ∪ F)
f) Are the events E and F mutually exclusive?
g) Are the events E and F complementary?
Example 2: Let’s consider the experiment of flipping a fair coin two times and observing the resulting sequence
of ”heads” and ”tails”.
a) Determine the sample space.
b) Find the event E where E = {x|x has one or more heads}
c) Find the event F where F = {x|x has more than 2 heads}
d) List all events of this experiment.
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Example 3: An experiment consists of casting a pair of dice and observing the number that falls uppermost on
each die.
(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)
S=
(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)
a) Determine the event that the sum of the numbers falling uppermost is less than or equal to 6.
b) Determine the event that the number falling uppermost on one die is a 4 and the number falling uppermost on
the other die is greater than 4.
Example 4: The manager of a local bank observes how long it takes a customer to complete his transactions at
the ATM.
a) Describe an appropriate sample space for this experiment.
b) Describe the event that it takes a customer between 2 and 3 minutes, inclusive, to complete his transactions at
the ATM.
Section 7.1 Highly Suggested Homework Problems: 3, 9, 11, 13, 19, 21, 23, 25, 27, 29, 31, 37, 39
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Section 7.2 - Definition of Probability
Definition: Suppose we repeat an experiment n times and an event E occurs m of those times. Then
the relative frequency of the event E.
m
n
is called
Example 1: Let’s say you flip a coin 100 times and a head occurs 61 times. What is the relative frequency of the
event E = {x|x is heads}?
Definition: Often, the more we repeat an experiment, the more the relative frequency approaches a certain value.
We call this the empirical probability of the event.
Definition: The probability of an event is a number between 0 and 1 that represents the likelihood of the event
occuring. The larger the probability, the more likely the event is to occur.
Definition: An event which consists of exactly one outcome is called a simple event of the experiment.
Example 2: List the simple events associated with each of the given experiments:
a) A nickel and a dime are tossed, and the result of heads or tails is recorded for each coin.
b) As part of a quality-control procedure, eight circuit boards are checked, and the number of defectives is
recorded.
Definition: The table that lists the probability of each simple event in an experiment is known as the probability
distribution.
Example 3: Metro Telephone Company compiled the accompanying information during a service-utilization
pertaining to the number of customers using their Dial-the-Time service from 7 A.M. to 9 A.M. on a certain
weekday morning. Using these data, find the probability distribution associated with the experiment.
Calls Received per Minute 10 11 12
Frequency of Occurrence
6 15 30
3
13 14 15
3 30 36
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Definition: The function which assigns a probability to each of the simple events is called a
probability function. It has the following properties:
1. 0 ≤ P(si ) ≤ 1
2. P(s1 ) + P(s2 ) + · · · + P(sn ) = 1
3. P({si } ∪ {s j }) = P(si ) + P(s j ) for i 6= j
Definition: Sample spaces in which the outcomes are equally likely are called uniform sample spaces. For a
uniform sample space S = {s1 , s2 , . . . , sn }, we can assign to the simple events s1 , s2 , . . . , sn the probabilities:
P(s1 ) = P(s2 ) = · · · = P(sn ) =
1
n
Finding the probability of an event E
1. Determine a sample space S associated with the experiment.
2. Assign probabilities to the simple events of S.
3. If E = {s1 , . . . , sm } where {s1 }, ...{sm } are simple events then
P(E) = P(s1 ) + P(s2 ) + · · · + P(sm )
4. If E is the empty set,0,
/ then P(E) = 0
Example 4: Let S = {s1 , s2 , s3 , s4 , s5 } be the sample space associated with an experiment having the following
probability distribution:
Outcome
Probability
s1
s2
s3
s4
s5
1
14
3
14
6
14
2
14
2
14
Find the probability of the event:
a) A = {s1 , s5 }
b) B = {s1 , s2 , s4 }
c) C = S
Example 5: If a marble is selected at random from a bowl containing three red, two white, and five black, what
is the probability that the marble drawn is not white?
Section 7.2 Highly Suggested Homework Problems: 3, 7, 9, 17, 23, 25, 27, 29, 47
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Section 7.3 - Rules of Probability
Properties:
1. P(E) ≥ 0 for any event E
2. P(S) = 1
3. If E and F are mutually exclusive (that is, E ∩ F = 0),
/ then
P(E ∪ F) = P(E) + P(F)
4. P(E ∪ F) = P(E) + P(F) − P(E ∩ F)
5. P(E c ) = 1 − P(E)
Example 1: Let E and F be two events of an experiment with sample space S. Suppose P(E) = 0.7, P(F) = 0.5,
and P(E ∩ F) = 0.3. Compute:
a) P(E ∪ F)
b) P(E c )
c) P(F c )
d) P(E c ∩ F c )
e) P(E c ∩ F)
Example 2: An experiment consists of selecting a card at random from a 52-card deck. What is the probability
that a diamond or a king is drawn?
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Example 3: A pair of fair six-sided dice is cast and the number that appears uppermost on each die is observed.
Determine the probability of the following events:
a) A double is thrown.
b) The sum of the numbers is at least 4.
c) At least one of the die is showing a 6.
Example 4: Among 500 freshman pursuing a business degree at a university, 320 are enrolled in an Economics
course, 225 are enrolled in a Mathematics course, and 140 are enrolled in both an Economics and a Mathematics
course. What is the probability that a freshman selected at random from this group is enrolled in:
a) an Economics or Mathematics course?
b) exactly one of these two courses?
c) neither an Economics course nor a Mathematics course?
Section 7.3 Highly Suggested Homework Problems: 1, 3, 5, 7, 9, 11, 13, 15, 19, 21, 23, 25, 27, 31, 35, 45
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Section 7.4 - Use of Counting Techniques in Probability
First, some counting problems from Sections 6.3 and 6.4:
Example 1: An investor has selected a mutual fund to invest his money in. He plans on observing its performance
over the next ten years. He will consider the year a success (S) if the mutal fund performs above average and a
failure (F) otherwise.
a) How many different outcomes are possible?
b) How many different outcomes have exactly six successes?
c) How many different outcomes have at least two successes?
Example 2: Five cards are randomly selected from a standard deck of 52 cards to form a poker hand. Determine
the number of ways a person can be dealt a full house (that is a three of a kind and a two of a kind)
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Computing the probability of an event in a uniform sample space: Let S be a uniform sample space and let
E be any event. Then,
n(E)
P(E) =
n(S)
where n(E) is the number of outcomes in E and n(S) is the number of outcomes in S.
Example 3: A sample of four marbles is selected from a bowl containing five white and eight green marbles.
Find the probability that at least two of the marbles are white.
Example 4: An unbiased coin is tossed six times. What is the probability that the coin will land heads
a) Exactly three times?
b) At most three times?
c) On the first and the last toss?
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Example 5: a) An exam consists of ten true-or-false questions. If a student randomly guesses on each question,
what is the probability that he or she will answer exactly six questions correctly?
b) An exam consists of ten multiple choice questions each having five choices of which only one is correct.
If a student randomly guesses on each question, what is the probability that he or she will answer exactly six
questions correctly?
Example 6: Two cards are selected at random without replacement from a well-shuffled deck of 52 playing cards.
Find the probability that two cards of the same suit are drawn.
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Example 7: Thirty people are selected at random.
a) What is the probability that none of the people in this group have the same birthday?
b) What is the probability that at least two people in this group have the same birthday?
Section 7.4 Highly Suggested Homework Problems: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 35, 37, 39, 41
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Section 7.5 - Conditional Probability and Independent Events
Example 1: A survey is done of people making purchases at a gas station:
buy drink (D) no drink (Dc )
buy gas (G)
20
15
no gas (Gc )
10
5
Total
30
20
Total
35
15
50
a) What is the probability that a person buys a drink?
b) What is the probability that a person doesn’t buy a drink?
c) What is the probability that a person buys gas and a drink?
d) What is the probability that a person buys gas but not a drink?
e) What is the probability that a person who buys a drink also buys gas?
f) What is the probability that a person who doesn’t buy a drink buys gas?
Definition: If E and F are events in an experiment and P(E) 6= 0, then the conditional probability that the event
F will occur given that the event E has already occurred is
Definition: The Product Rule is found by rearranging the above formula as follows:
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Example 2: Let’s use a tree diagram to help us understand the product rule:
Example 3: At a party, 1/3 of the guest are women. Seventy-five percent of the women wore sandals and 40%
of the men wore sandals.
a) What is the probability that a person chosen at random at the party is a man wearing sandals?
b) What is the probability that a person chosen at random is wearing sandals?
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Example 4: Consider drawing 3 cards from a standard deck of 52 cards without replacement.
a) What is the probability that the 3 cards are hearts?
b) What is the probability that the third card drawn is a heart given the first two cards are hearts?
Definition: If A and B are independent events, then
and B, are independent if and only if
and
. Thus, two events, A
Example 5: A pair of fair six-sided dice is rolled. Let E be the event that the sum of the numbers landing
uppermost is seven and let F be the event that exactly one four is rolled. Are the events E and F independent?
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Example 6: Meghan and Natalie go to Freebirds. After ordering their food they get to roll a pair of fair dice for
a chance to get their meal for free. Each die has six sides with one of the sides having a backwards F on it. If
both of the dice land on the backwards F, you win. What is the probability that at least one of the girls wins?
Section 7.5 Highly Suggested Homework Problems: 1, 3, 5, 7, 9, 13, 15, 17, 19, 21, 23, 25, 30, 33, 35, 37, 41
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Section 7.6 - Bayes’ Theorem
Example 1: If we are given information about P(F|E), can we find P(E|F)?
Definition: The above formula is known as Bayes’ Theorem.
Example 2: We are to choose a marble from a cup or a bowl. We flip a fair coin to decide whether to choose
from the cup or the bowl. The bowl contains 1 red and 2 green marbles. The cup contains 3 red and 2 green
marbles. What is the probability that a marble came from the bowl given that it is red?
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Example 3: A crate contains 7 basketballs and 4 footballs. A bag contains 4 basketballs and 2 footballs. A ball
is drawn at random from the crate and put in the bag. A ball is then drawn from the bag. Given that a basketball
was chosen from the bag, what is the probability that a football was drawn from the crate?
Example 4: Two cards are drawn in succession without replacement from a standard deck of 52 cards. What is
the probability that the first card is a face card given that the second card is an ace?
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Example 5: Complete the following tree diagram and use it to answer the following questions:
a) Find P(E).
b) Find P(A ∪ D).
c) Find P(B|E).
d) Are E and B independent events?
Section 7.6 Highly Suggested Homework Problems: 7, 9, 13, 15, 17, 19, 21, 33, 42, 50
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