1. Lial/Hungerford/Holcomb/Mullins: Mathematics with Applications 11e Finite Mathematics with Applications 11e
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2. Counting, Probability Distributions, and Further Topics in Probability
Chapter 9
Counting, Probability
Distributions, and Further Topics in Probability
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3. Probability Distributions and Expected Values
Section 9.1
Probability Distributions and Expected Values
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Example:
Each day, Lynette and Tanisha toss a coin to see who buys coffee (at $1.75 a cup). One tosses, while the other calls the outcome. If the person who calls the outcome is correct, the other buys the coffee; otherwise the caller pays. Find Lynette’s expected winnings.
Solution:
Assume that an honest coin is used, that Tanisha tosses that coin and that Lynette calls the outcome. The possible results and corresponding probabilities are shown in the following table:
Lynette wins a $1.75 cup of coffee whenever the results and calls match, and she loses $1.75 when there is no match. Her expected winnings are
On the average, over the long run, Lynette breaks even.
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11. The Multiplication Principle, Permutation, and Combinations
Section 9.2
The Multiplication Principle, Permutation, and Combinations
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Example:
Early in 2012, 7 candidates sought the Republican nomination for president at the Iowa caucus. In a poll, how many ways could voters rank their first, second, and third choices?
Solution:
This is the same as finding the number of permutations of 7 elements taken 3 at a time. Since there are 3 choices to be made, the multiplication principle gives Alternatively, by the formula for
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25. Applications of Counting
Section 9.3
Applications of Counting
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Example:
In a common form of 5-card draw poker, a hand of 5 cards is dealt to each player from a deck of 52 cards. There is a total of such hands possible. Find the probability of being dealt each of the given hands.
(a) Heart-flush hand (5 hearts)
Solution:
There are 13 hearts in a deck; there are
different hands containing only hearts. The probability of a flush is
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Example:
In a common form of 5-card draw poker, a hand of 5 cards is dealt to each player from a deck of 52 cards. There is a total of such hands possible. Find the probability of being dealt each of the given hands.
(b) A flush of any suit (5 cards, all from 1 suit)
Solution:
There are 4 suits to a deck, so
(c) A full house of aces and eights (3 aces and 2 eights)
Solution:
There are ways to choose 3 aces from among the 4 in the deck and
ways to choose 2 eights, so
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Example:
In a common form of 5-card draw poker, a hand of 5 cards is dealt to each player from a deck of 52 cards. There is a total of such hands possible. Find the probability of being dealt each of the given hands.
(d) Any full house (3 cards of one value, 2 of another)
Solution:
There are 13 values in a deck, so there are 13 choices for the first value mentioned, leaving 12 choices for the second value. (Order is important here, since a full house of aces and eights, for example is not the same as a full house of eights and aces.)
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Section 9.4
Binomial Probability
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31. Suppose a family has 3 children.
Example:
Suppose a family has 3 children.
(a) Find the probability distribution for the number of girls.
Solution:
Let x = the number of girls in three births. According to the binomial probability rule, the probability of exactly one girl being born is
The other probabilities in this distribution are found similarly, as shown in the following table:
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32. Suppose a family has 3 children.
Example:
Suppose a family has 3 children.
(b) Find the expected number of girls in a 3-child family.
Solution:
For a binomial distribution, we can use the following method (which is presented here with a “plausibility argument,” but not a full proof): Because 50% of births are girls, it is reasonable to expect that 50% of a sample of children will be girls. Since 50% of 3 is we conclude that the expected number of girls
is 1.5.
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Section 9.5
Markov Chains
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41. Decide whether the given transition matrices are regular.
Example:
Decide whether the given transition matrices are regular.
(a)
Solution:
Square A:
Since all entries in A2 are positive, matrix A is regular.
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42. Decide whether the given transition matrices are regular.
Example:
Decide whether the given transition matrices are regular.
(b)
Solution:
Find various powers of B:
Notice that all of the powers of B shown here have zeros in the same locations. Thus, further powers of B will still have the same zero entries, so that no powers of matrix B contains all positive entries. For this reason, B is not regular.
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Section 9.6
Decision Making
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Example:
An owner of several greeting-card stores must decide in July about the type of displays to emphasize for Sweetest Day in October. He has three possible choices: emphasize chocolates, emphasize collectible gifts, or emphasize gifts that can be engraved. His success is dependent on the state of the economy in October. If the economy is strong, he will do well with the collectible gifts, while in a weak economy, the chocolates do very well. In a mixed economy, the gifts that can be engraved will do well. He first prepares a payoff matrix for all three possibilities, where the numbers in the matrix represent his profits in thousands of dollars:
Use the payoff matrix to answer the following questions.
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Example:
Use the payoff matrix to answer the following questions.
(a) What would an optimist do?
Solution:
If the owner is an optimist, he should aim for the biggest number on the matrix, 110 (representing $110,000 in profit). His strategy in this case would be to display collectibles.
(b) How would a pessimist react?
Solution:
A pessimist wants to find the best of the worst things that can happen. If he displays collectibles, the worst that can happen is a profit of $45,000. For displaying engravable items, the worst is a profit of $60,000, and for displaying chocolates, the worst is a profit of $30,000. His strategy here is to use the engravable items.
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Example:
Use the payoff matrix to answer the following questions.
(c)
Suppose the owner reads in a business magazine that leading experts believe that there is a 50% change of a weak economy in October, a 20% chance of a mixed economy, and a 30% chance of a strong economy. How might he use this information?
Solution:
The owner can now find his expected profit for each possible strategy.
Here, the best strategy is to display gifts that can be engraved; the expected profit is 74.5, or $74,500.
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