Section 8.2-8.4
Notes for math 141
Finite Mathematics
8.2: Expected Value
Example 1: Records kept by the chef dietitian at the university cafeteria over a
25-wk period show the following weekly consumption of milk (in gallons):
Milk
Weeks
200 201 202 203
4
6
8
5
204
2
Find the average number of gallons of milk consumed per week in the cafeteria.
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Notes for math 141
Section 8.2-8.4
Finite Mathematics
Definition: Let X denote a random variable that assumes the values x1 , x2 ⋯xn ,
with associated probabilities p1 , p2 , pn , respectively. Then the expected value of X,
E(X), is given by:
E(X) = x1 p1 + x2 p2 + ⋯xn pn
Example 2: Referring to Example 1, let the random variable X denote the number
of gallons of milk consumed in a week at the cafeteria.
a) Find the probability distribution of X.
b) Compute E(X)
c) Draw a histogram for the random variable X
Example 3:In a lottery, 3000 tickets are sold for $1 each. One first prize of $5000, 1
second prize of $1000, 3 third prizes of $100, and 10 consolation prizes of $10 are to
be awarded. What are the expected net earnings of a person who buys one ticket?
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Notes for math 141
Section 8.2-8.4
Finite Mathematics
Example 4: A woman purchased a $15,000 1-year term-life insurance policy for
$200. Assuming that the probability that she will live another year is 0.994, find the
companys expected net gain.
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Notes for math 141
Section 8.2-8.4
Finite Mathematics
Definition: A fair game means that the expected value of the players net winnings
is zero. (E(X) = 0)
Example 5: Mike and Bill play a card game with a standard deck of 52 cards.
Mike selects a card from a well-shuffled deck and receives A dollars from Bill if the
card selected is a diamond; otherwise, Mike pays Bill a dollar. Determine the value
of A if the game is to be fair.
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Notes for math 141
Section 8.2-8.4
Finite Mathematics
Definition: If P(E) is the probability of an event E occuring, then a) The odds in
favor of E occurring are
P (E)
P (E)
=
, P (E c ) ≠ 0
1 − P (E) P (E c )
b) The odds against E occurring are
1 − P (E) P (E c )
=
, P (E) ≠ 0
P (E)
P (E)
Example 6: The probability of an event E occurring is .8.
a) What are the odds in favor of E occurring?
b) What are the odds against E occurring?
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Notes for math 141
Section 8.2-8.4
Finite Mathematics
Definition: If the odds in favor of an event E occurring are a to b, then the probability of E occurring is
a
P (E) =
.
a+b
Example 7: If a sports forecaster states that the odds of a certain boxer winning a
match are 4 to 3, what is the probability that the boxer will win the match?
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Notes for math 141
Section 8.2-8.4
Finite Mathematics
Definition: The median is the middle value in a set of data arranged in increasing
or decreasing order (when there is an odd number of entries). If there is an even
number of entries, the median is the average of the two middle numbers.
Definition: The mode is the value that occurs most frequently in a set of data.
Example 7: In each of the following cases, find the mean, median, and mode.
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Notes for math 141
Section 8.2-8.4
Finite Mathematics
7.7 Variance and Standard Deviation
Example 8: Draw the histograms for the random variables X and Y that have the
following probability distributions:
x P(X=x)
1
.1
2
.1
3
.6
4
.1
5
.1
y P(Y=y)
1
.3
2
.1
3
.2
4
.1
5
.3
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Notes for math 141
Section 8.2-8.4
Finite Mathematics
Definition: Let X denote a random variable that assumes the values x1 , x2 ⋯xn ,
with associated probabilities p1 , p2 , pn , respectively, and the expected value E(X) = µ.
Then the variance of the random variable X is
V ar(X) = p1 (x1 − µ)2 + p2 (x2 − µ)2 + ⋯ + pn (xn − µ)2
.
Definition: The standard deviation of a random variable X, σ, is defined by:
√
σ = V ar(X).
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Notes for math 141
Section 8.2-8.4
Finite Mathematics
Finding expected value and standard deviation using the calculator:
• Enter the x-values into L1 and the corresponding probabilities into L2 (ST AT →
1:Edit)
• On the homescreen type 1-Var Stats L1 , L2 (STAT→CALC→1:1-Var Stats)
• x̄ is the expected value (mean)
• σx is the standard deviation
• To find variance, you would need to recall that V ar(X) = σ 2
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Notes for math 141
Section 8.2-8.4
Finite Mathematics
Example 9: Referring to Example 8,
a) Find the variance and standard deviation of X.
b) Find the variance and standard deviation of Y.
Example 10: If you roll a fair six-sided die 4 times, what is the variance and standard
deviation of the number of times the die lands on a two?
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Notes for math 141
Section 8.2-8.4
Finite Mathematics
8.4: The binomial distribution
Definition: Definition: A bernoulli trial is an experiment in which the outcome
can be either of two possible outcomes: success and failure.
Definition: A binomial experiment (OR bernoulli process OR repeated bernoulli trials) is just a repetition of many bernoulli trials. A binomial experiment has the
following properties:
• The number of trials in the experiment is fixed.
• There are 2 possible outcomes in each trial: success and failure
• The probability of success in each trial is the same.
• Trials are independent of each other.
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Notes for math 141
Section 8.2-8.4
Finite Mathematics
Calculating a Binomial Probability:
• . Determine if the experiment is binomial.
• Determine the number of trials (n).
• Define success in the experiment and determine the probability of that success
occurring (p).
• Determine the number of successes desired (r).
• Calculate the desired probability:
(a) By hand, the probability is found by doing the following calculation:
(b) We can use the calculator functions binompdf ( or binomcdf to do these calculations for us).
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Notes for math 141
Section 8.2-8.4
Finite Mathematics
Example 11: A fair die is cast 4 times. Compute the probability of obtaining exactly
one 6 in the four throws.
Example 12:A biology quiz consists of 8 multiple choice questions. Each question has five choices of which only one is correct. If a student randomly guesses on
each question, what is the probability of
a) getting at most 3 questions right?
b) getting at least 2 questions right?
c) getting more than 2 but less than 6 questions right?
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Notes for math 141
Section 8.2-8.4
Finite Mathematics
Example 13: Studies have shown that 30selected, what is the probability that at
least 18 but fewer than 35 of them were delivered by C-section?.
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Notes for math 141
Section 8.2-8.4
Finite Mathematics
Definition: The variance of a binomial distribution with n trials and the probability
of success equal to p is
Example 14: If you roll a fair six-sided die 100 times, what is the variance and
standard deviation of the number of times that the die will land on a number less
than three?
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