Notes for math 141
Section 7.2-7.4
Finite Mathematics
7.2: Definition of probability
Definition: Suppose that in n trails an event E occurs m times , we call
frequency of the event E after n repititions.
m
n
a relative
Example 1: Let us 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 a head}.
Definition: If this relative frequency approaches some value P (E), then P (E) is
called the Empirical Probability of E.
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.
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Notes for math 141
Section 7.2-7.4
Finite Mathematics
Definition: Let S be a finite sample space with n outcomes, then the events consist
of exactly one point are called the simple events of the experiment.
Example 2: List the simple events of each experiments.
a. Consider the experiment of flipping a fair coin two times and observing the resulting sequence of heads and tails.
b. Consider the experiment of rolling a fair six-sided die and observing the number that lands uppermost.
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Notes for math 141
Section 7.2-7.4
Finite Mathematics
Definition: The table that lists the probability of each simple event in an experiment
is known as probability distribution.
Example 3: An unfair 6-sided die is rolled 200 times and the number rolled each
time is recorded. The results are given below.
Number
Frequency
1 2
25 20
3 4 5 6
45 50 36 24
Find the probability distribution of the experiment.
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Notes for math 141
Section 7.2-7.4
Finite Mathematics
Definition: The function which assigns a probability to each simple event is called
a probability function. It has the following properties.
• 0 ≤ P (si ) ≤ 1, i = 1, ⋯, n
• P (s1 ) + ⋯ + P (sn ) = 1
• P ({si } ∪ {sj }) = P (si ) + P (sj ) (i ≠ j)
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Notes for math 141
Section 7.2-7.4
Finite Mathematics
Definition: A sample space S = {s1 , ⋯, sn } in which all of the outcomes are equally
likely is known as a uniform sample space. We know that P (s1 ) = ⋯ = P (sn ) = n1 .
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 }, then P (E) = P (s1 ) + ⋯ + P (sm ), if E = ∅, then P (E) = 0
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Notes for math 141
Section 7.2-7.4
Finite Mathematics
Example 4: Suppose a single card is randomly drawn from a standard 52 card deck.
Determine the probability of each of the following events:
a) A king is drawn.
b) A heart is drawn.
Example 5: Suppose a fair six-sided die is rolled and the number that lands uppermost is observed.
a) Find the sample space for this experiment.
b) Find the probability that an odd number is rolled.
c) Find the probability that a 9 is rolled.
d) Find the probability that a number less than 8 is rolled.
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Notes for math 141
Section 7.2-7.4
Finite Mathematics
7.3: Rules of Propability
Property:
1. P (E) ≥ 0for any event E
2. P (S) = 1
3. If E and F are mutually exclusive (that is, E∩F = ∅), 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)
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Notes for math 141
Section 7.2-7.4
Finite Mathematics
Example 6: 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 7: 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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Notes for math 141
Section 7.2-7.4
Finite Mathematics
Example 8: 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?
Example 9:A salesman always makes a sale at at least one of the three stops in
Atlanta. He makes a sale at only the first stop 30% of the time, 15% at only the
second stop, and 20% at only the third stop. It was also found that he makes a sale
at exactly two of the stops 25% of the time. Find the probability that the salesman
makes a sale at all three stops in Atlanta.
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Notes for math 141
Section 7.2-7.4
Finite Mathematics
7.4 Use of Counting Techniques in Probability
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
P (E) =
n(E)
,
n(S)
where n(E) is the number of outcomes in E and n(S) is the number of outcomes in
S.
Example 10: Four marbles are selected at random without replacement from a bowl
containing five white and eight green marbles. Find the probability that at least two
of the marbles are white.
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Notes for math 141
Section 7.2-7.4
Finite Mathematics
Example 11: 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?
Example 12:
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?
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Notes for math 141
Section 7.2-7.4
Finite Mathematics
Example 13: Two cards are selected at random without replacement from a wellshuffled deck of 52 playing cards. Find the probability that two cards of the same
suit are drawn.
Example 14: Five cards are randomly selected from a standard deck of 52 cards
to form a poker hand. Determine the probability that at least two of them are red,
Example 15: 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?
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