EXERCISES Section 2.3 (29-44)

29

As of April 2006, roughly 50 million .com web domain names were registered (e.g., yahoo.com).

1. How many domain names consisting of just two letters in sequence can be formed? How many domain names of length two are there if digits as well as letters are permitted as characters?
   • Note: A character length of three or more is now mandated.
2. How many domain names are there consisting of three letters in sequence? How many of this length are there if either letters or digits are permitted?
   • Note: All are currently taken.
3. Answer the questions posed in (b) for four-character sequences.
4. As of April 2006, 97,786 of the four-character sequences using either letters or digits had not yet been claimed. If a four-character name is randomly selected, what is the probability that it is already owned?

30

A friend of mine is giving a dinner party. His current wine supply includes

• 8 bottles of zinfandel,
• 10 of merlot,
• 12 of cabernet (he only drinks red wine),
• all from different wineries. Questions:

1. If he wants to serve 3 bottles of zinfandel and serving order is important, how many ways are there to do this?
2. If 6 bottles of wine are to be randomly selected from the 30 for serving, how many ways are there to do this?
3. If 6 bottles are randomly selected, how many ways are there to obtain two bottles of each variety?
4. If 6 bottles are randomly selected, what is the probability that this results in two bottles of each variety being chosen?
5. If 6 bottles are randomly selected, what is the probability that all of them are the same variety?

31

The composer Beethoven wrote 9 symphonies, 5 piano concertos (music for piano and orchestra), and 32 piano sonatas (music for solo piano).

1. How many ways are there to play first a Beethoven symphony and then a Beethoven piano concerto?
2. The manager of a radio station decides that on each successive evening (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?

32

An electronics store is offering a special price on a complete set of components (receiver, compact disc player, speakers, turntable). A purchaser is offered a choice of manufacturer for each component:

• Receiver: Kenwood, Onkyo, Pioneer, Sony, Sherwood
• Compact disc player: Onkyo, Pioneer, Sony, Technics
• Speakers: Boston, Infinity, Polk
• Turntable: Onkyo, Sony, Teac, Technics

A switchboard display in the store allows a customer to hook together any selection of components (consisting of one of each type). Use the product rules to answer the following questions:

1. In how many ways can one component of each type be selected?
2. In how many ways can components be selected if both the receiver and the compact disc player are to be Sony?
3. In how many ways can components be selected if none is to be Sony?
4. In how many ways can a selection be made if at least one Sony component is to be included?
5. If someone flips switches on the selection in a completely random fashion, what is the probability that the system selected contains at least one Sony component? Exactly one Sony component?

33

Again consider a Little League team that has 15 players on its roster.

1. How many ways are there to select 9 players for the starting lineup?
2. How many ways are there to select 9 players for the starting lineup and a batting order for the 9 starters?
3. Suppose 5 of the 15 players are left-handed. How many ways are there to select 3 left-handed outfielders and have all 6 other positions occupied by right-handed players?

34

Computer keyboard failures can be attributed to electrical defects or mechanical defects. A repair facility currently has 25 failed keyboards, 6 of which have electrical defects and 19 of which have mechanical defects.

1. How many ways are there to randomly select 5 of these keyboards for a thorough inspection (without regard to order)?
2. In how many ways can a sample of 5 keyboards be selected so that exactly two have an electrical defect?
3. If a sample of 5 keyboards is randomly selected, what is the probability that at least 4 of these will have a mechanical defect?

35

A production facility employs 10 workers on the day shift, 8 workers on the swing shift, and 6 workers on the graveyard shift. A quality control consultant is to select 5 of these workers for in-depth interviews. Suppose the selection is made in such a way that any particular group of 5 workers has the same chance of being selected as does any other group (drawing 5 slips without replacement from among 24).

1. How many selections result in all 5 workers coming from the day shift? What is the probability that all 5 selected workers will be from the day shift?
2. What is the probability that all 5 selected workers will be from the same shift?
3. What is the probability that at least two different shifts will be represented among the selected workers?
4. What is the probability that at least one of the shifts will be unrepresented in the sample of workers?

36

An academic department with five faculty members narrowed its choice for department head to either candidate $A$ or candidate $B$ . Each member then voted on a slip of paper for one of the candidates. Suppose there are actually three votes for $A$ and two for $B$ . If the slips are selected for tallying in random order, what is the probability that $A$ remains ahead of $B$ throughout the vote count (e.g., this event occurs if the selected ordering is $AABAB$ , but not for $ABBAA)$ ?

37

An experimenter is studying the effects of temperature, pressure, and type of catalyst on yield from a certain chemical reaction. Three different temperatures, four different pressures, and five different catalysts are under consideration.

1. If any particular experimental run involves the use of a single temperature, pressure, and catalyst, how many experimental runs are possible?
2. How many experimental runs are there that involve use of the lowest temperature and two lowest pressures?
3. Suppose that five different experimental runs are to be made on the first day of experimentation. If the five are randomly selected from among all the possibilities, so that any group of five has the same probability of selection, what is the probability that a different catalyst is used on each run?

38

A sonnet is a 14-line poem in which certain rhyming patterns are followed. The writer Raymond Queneau published a book containing just 10 sonnets, each on a different page. However, these were structured such that other sonnets could be created as follows: the first line of a sonnet could come from the first line on any of the 10 pages, the second line could come from the second line on any of the 10 pages, and so on (successive lines were perforated for this purpose).

1. How many sonnets can be created from the 10 in the book?
2. If one of the sonnets counted in part (a) is selected at random, what is the probability that none of its lines came from either the first or the last sonnet in the book?

39

A box in a supply room contains 15 compact fluorescent lightbulbs, of which 5 are rated 13-watt, 6 are rated 18-watt, and 4 are rated 23-watt. Suppose that three of these bulbs are randomly selected.

1. What is the probability that exactly two of the selected bulbs are rated 23-watt?
2. What is the probability that all three of the bulbs have the same rating?
3. What is the probability that one bulb of each type is selected?
4. If bulbs are selected one by one until a 23-watt bulb is obtained, what is the probability that it is necessary to examine at least 6 bulbs?

40

Three molecules of type , three of type , three of type, and three of type $D$ are to be linked together to form a chain molecule. One such chain molecule is $ABCDABCDABCD$ , and another is $BCDDAAABDBCC$ .

1. How many such chain molecules are there?
   • Hint: If the three $A$ ‘s were distinguishable from one another - $A_1,A_2,A_3$ - and the $B$ ‘s, $C$ ‘s, and $D$ ‘s were also, how many molecules would there be? How is this number reduced when the subscripts are removed from the A’s?
2. Suppose a chain molecule of the type described is randomly selected. What is the probability that all three molecules of each type end up next to one another (such as in BBBAAADDDCCC)?

41

An ATM personal identification number (PIN) consists of four digits, each a $0,1,2,\dots 8$ , or 9, in succession.

1. How many different possible PINs are there if there are no restrictions on the choice of digits?
2. According to a representative at the author’s local branch of Chase Bank, there are in fact restrictions on the choice of digits. The following choices are prohibited:
   1. all four digits identical
   2. sequences of consecutive ascending or descending digits, such as 6543
   3. any sequence starting with 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)?
3. Someone has stolen an ATM card and knows that the first and last digits of the PIN are 8 and 1, respectively. He has three tries before the card is retained by the ATM (but does not realize that). So he randomly selects the $2^{\text{nd }}$ and $3^{\text{rd }}$ 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?
4. Recalculate the probability in (c) if the first and last digits are 1 and 1 , respectively.

42

A starting lineup in basketball consists of two guards, two forwards, and a center.

1. A certain college team has on its roster three centers, four guards, four forwards, and one individual (X) who can play either guard or forward. How many different starting lineups can be created?
   • Hint: Consider lineups without $\mathrm{X}$ , then lineups with $\mathrm{X}$ as guard, then lineups with $\mathrm{X}$ as forward.
2. Now suppose the roster has 5 guards, 5 forwards, 3 centers, and 2 “swing players” (X and Y) who can play either guard or forward. If 5 of the 15 players are randomly selected, what is the probability that they constitute a legitimate starting lineup?

43

In five-card poker, a straight consists of five cards with adjacent denominations (e.g., 9 of clubs, 10 of hearts, jack of hearts, queen of spades, and king of clubs). Assuming that aces can be high or low, if you are dealt a five-card hand, what is the probability that it will be a straight with high card 10 ? What is the probability that it will be a straight? What is the probability that it will be a straight flush (all cards in the same suit)?

44

Show that $\left(\begin{array}{l}n\\ k\end{array}\right)=\left(\begin{array}{c}n\\ n-k\end{array}\right).$ Give an interpretation involving subsets.