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Chapter 2: Q44E (page 75)
Show that \(\left( {\begin{array}{*{20}{l}}{\rm{n}}\\{\rm{k}}\end{array}} \right){\rm{ = }}\left( {\begin{array}{*{20}{c}}{\rm{n}}\\{{\rm{n - k}}}\end{array}} \right)\). Give an interpretation involving subsets.
There are same number of subsets of those sizes.
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Human visual inspection of solder joints on printed circuit boards can be very subjective. Part of the problem stems from the numerous types of solder defects (e.g., pad non wetting, knee visibility, voids) and even the degree to which a joint possesses one or more of these defects. Consequently, even highly trained inspectors can disagree on the disposition of a particular joint. In one batch of 10,000 joints, inspector A found 724 that were judged defective, inspector B found 751 such joints, and 1159 of the joints were judged defective by at least one of the inspectors. Suppose that one of the 10,000 joints is randomly selected.
a. What is the probability that the selected joint was judged to be defective by neither of the two inspectors?
b. What is the probability that the selected joint was judged to be defective by inspector B but not by inspector A?
The composer Beethoven wrote \({\rm{9}}\) symphonies, \({\rm{9}}\) piano concertos (music for piano and orchestra), and \({\rm{32}}\) piano sonatas (music for solo piano).
a. How many ways are there to play first a Beethoven symphony and then a Beethoven piano concerto?
b. The manager of a radio station decides that on each successive evening (\({\rm{7}}\) days per week), a Beethoven symphony will be played followed by a Beethoven piano concerto followed by a Beethoven piano sonata. For how many years could this policy be continued before exactly the same program would have to be repeated?
An ATM personal identification number (PIN) consists of four digits, each a\({\rm{0, 1, 2, \ldots 8, or 9}}\), in succession.
a. How many different possible PINs are there if there are no restrictions on the choice of digits?
b. According to a representative at the author鈥檚 local branch of Chase Bank, there are in fact restrictions on the choice of digits. The following choices are prohibited: (i) all four digits identical (ii) sequences of consecutive ascending or descending digits, such as \({\rm{6543}}\) (iii) any sequence starting with \({\rm{19}}\) (birth years are too easy to guess). So, if one of the PINs in (a) is randomly selected, what is the probability that it will be a legitimate PIN (that is, not be one of the prohibited sequences)?
c. Someone has stolen an ATM card and knows that the first and last digits of the PIN are \({\rm{8}}\) and\({\rm{1}}\), respectively. He has three tries before the card is retained by the ATM (but does not realize that). So, he randomly selects the \({\rm{2nd and 3rd}}\) digits for the first try, then randomly selects a different pair of digits for the second try, and yet another randomly selected pair of digits for the third try (the individual knows about the restrictions described in (b) so selects only from the legitimate possibilities). What is the probability that the individual gains access to the account?
d. Recalculate the probability in (c) if the first and last digits are \({\rm{1}}\) and \({\rm{1}}\), respectively.
Consider the system of components connected as in the accompanying picture. Components \({\rm{1}}\) and \({\rm{2}}\) are connected in parallel, so that subsystem works iff either 1 or 2 works; since \({\rm{3}}\)and\({\rm{4}}\) are connected in series, that subsystem works iff both\({\rm{3}}\)and\({\rm{4}}\)work. If components work independently of one another and P(component i works) \({\rm{ = }}{\rm{.9}}\)for \({\rm{i = }}{\rm{.1,2}}\)and \({\rm{ = }}{\rm{.8}}\)for \({\rm{i = 3,4}}\),calculate P(system works).
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