from file folder COMBO.131 (“finite math” problems)

Exam 2. August 23, 1999

1.
A club with 12 members elects a President, a Vice-President, a Deputy Vice-President, and a Go-fer (no member can serve in two different offices). How many different election results are possible?

2.
A student believes that one-fourth of the 40 multiple-choice problems on a certain exam are to be answered “A”; one-fourth are to be answered “B”; one-fourth “C”; and one-fourth “D”. Give a formula for the number of possible patters of answers meeting these conditions. Hint: consider the 40-letter “word”
AAAAAAAAAABBBBBBBBBBCCCCCCCCCCDDDDDDDDDD

3.
An “outfit” for the day consists of a choice of T-shirt, a pair of jeans, and a pair of shoes. I have 30 T-shirts, 4 pairs of jeans, and 2 pairs of shoes to choose from. How many different “outfits” are possible?

4.
Write down the sample space (16 outcomes) for the experiment “toss a coin four times”.
a.
What is the probability that exactly two heads occur?
b.
What is the probability that at least two heads occur?

5.
I have 6 math books, 3 novels, and 1 Bible on my coffeetable. I decide to put four of these books in my backpack.
a.
How many different sets of four books do I have to choose from?
b.
How many of these do not include the Bible?
c.
How many include exactly one novel?

6.
A “Texas Hold’em” hand is a set of two cards chosen from a standard deck. How many different Hold’em hands are there?

How many Hold’em hands are “high pairs”—defined here as a pair of Jacks, Queens, Kings, or Aces?

7.
How many distinguishable arrangements are there for the letters of “ELEEMOSYNARY”? (Bonus Point: what does this word mean?)

8.
A billionaire awards large cash prizes to 5 lucky banjo players. On Monday, he will give away 1 million dollars; on Tuesday, 2 million; and so on until Friday, when he will give away 5 million. There are 11 banjo players on his short-list. In how many ways can the prizes be awarded?

Quiz. February 21, 2002

1.
For the sample space S={h,a,d,n,o,e,x,i,t,s}
and events E = {d,e,a,t,h} and F={t,a,x,e,s} determine the indicated events
a. E’
b. F’
c. EuF
d. (EuF)’
e. E’uF’
f. E&F
g. E&F’
h. (E&F)’

2.
Suppose P(E)=0.3, P(F)=0.4, and P(E&F)=0.2. Find a. P(E’) and b. P(EuF).

3.
A couple has four children. Write out all (sixteen) of the possible birth orders: BBBB (four boys), BBBG,…,GGGG. What is the probability that there are more girls than boys?

4.
A poker hand is a set of five cards chosen randomly from the standard deck. A flush is five cards all of the same suit. Find the probability of a flush (hint: four times the probability of a diamond flush).

5.
How many distinguishable arrangements are there for the letters of PARAMETER?

6.
How many (seven digit) phone numbers can be created without using a “0”?

How many of them have no repeated digit?

7.
A gunsmith has 4 locks, 5 stocks, and 6 barrels. He selects one of each. In how many ways can he make his selection?

Advertisements

  1. My questions:

    Question 5 and (we can distinguish 2 novels, no?)

    Quiz Question 4 (is a straight-flush a flush?)

  2. 5. we can, yes.

    thus
    *********************************************
    S = {
    m_1,m_2,m_3,m_4,m_5,m_6,
    n_1,n_2,n_3,
    B
    }
    is given.
    how many (of the 2^10 = 1K)
    subsets (X) of S
    have the property that
    a.) X has four “points”.

    \*comment: in our offices with the doors
    \*comment: closed, graduates of math
    \*comment: departments refer to “points”
    \*comment: of a “space” more or less
    \*comment: interchangeably with “elements”
    \*comment: of a “set”

    #X = 4
    in other words.
    (a handy abbreviation.)

    b.) #X=4 and B\not\in X
    c.) #X=4 and X\intersect {n_1, n_2, n_3}\not\equal\nullset
    ********************************************
    is an elaborate rephrasing of 5.)

    on the much more subtle question
    “is a straight flush a flush”.

    first of all: good question.
    now.

    my immediate reaction:
    i read the problem again.
    “a flush is five cards of
    the same suit.”
    i imagine myself in the position
    of a reader knowing nothing of
    the rules of poker…
    but knowing the suits and ranks
    of the 4 by 13 array beloved
    of all teachers of probability…
    and i take this definition as i
    more or less *must*:
    inclusive of *all*
    sets-of-five having
    no second suit.

    there’s a great deal hidden here.
    the teacher hopes the student
    knows the deck… we’ve practiced
    some in class and whatnot.
    the student hopes the teacher
    isn’t unconsciously *loading* the
    deck (again) with culturally-
    -biased assumptions of which
    they may only be very dimly
    aware.

    “extra credit”… very likely just
    a minute of fuss made in class by a
    widely grinning instructor…
    for a student just for
    (coherently) asking this question.

    i’m looking for
    4\binom{13}{5}
    in other words.
    considering straight-flushes
    as *excluded* is a harder
    question… of the type i’d
    never have dared put on
    a quiz for classes like “131”.
    maybe as a take-home.

    is a royal flush a straight flush?
    does the buddha have a cat nature?

  3. As always, when I introduce cards, I slowly run through suits, colors, ranks, face vs number, etc, etc. “You probably know all of this, but be patient, I am certain that there are 2 or 3 of you for whom this is not familiar information” (2 or 3, not 1 or 2, because if 1, and you don’t know, then I’ve singled you out. With 2 or 3, if you don’t know, you have a secret compadre).

    Anyhow, when the kiddies write their course evaluations (I have teased out fairly honest, fairly thoughtful, short answer type questions), this year I got a complaint about not teaching the cards well enough for those who don’t know them well. Hmmmph. I need to be more careful.

    For the non-player, all flushes are flushes. For the player, straight-flushes are not flushes, and royal flushes are simply the highest ranked straight-flushes.

  4. A student, a player, and a math major, thinks I’m wrong (he didn’t say it like THAT, in fact, I posed the question roundabout), but it’s enough for me to want to back off the flushes…

  5. vlorbik

    we like to use language like
    “any straight flush beats any flush”…
    but in a math-class setting it’s best
    in my opinion to regard this as *slang*.

    more evidence not that any is needed
    that “real life” problems are *harder*
    than “abstract” problems.

    “real life” definitions are *always* messy
    (because nothing *is* what it *seems*).
    *only* in the realm of the purely symbolic
    can we ever actually mean *exactly* what
    we say…

  6. Indeed. A pair of Jacks is better than nothing, and nothing beats a straight-flush…

    Covering my error with a cute exit.

  7. vlorbik

    answers and remarks

    1)
    P(12,4)=12.11.10.9=11880

    the middle one is hugely
    to be preferred if one is
    allowed only *one*…

    and is not provided a context.

    which is absurd i hope but bear
    with me for purposes of argument.

    because nobody ever wants to believe
    contexts matter or something and if
    i can’t put some brief stop to it in a
    comment on my own post in my own
    blog well i’m in even more trouble
    than i think i am. (which, as you can
    imagine, gets *tiring* when it keeps
    on *happening* over and over.)

    the middle one, then,
    12.11.10.9,
    sort of *tells its own story*.

    it tells the story much more clearly
    to well-prepared minds, of course,
    like any story… on parle francais ici?
    me neither, but you get the idea. so.
    what’s this *preparation*?

    in my work, nominally, it would consist,
    minimally, of having seen a lecturer
    go through at least one example
    a great deal *like* it.

    the “big picture” one hopes to have
    instilled with this work consists,
    for problems like this, of the idea
    that one has here a *sequence*
    of things we can think of as
    occurring in over *time*;
    that one can then divide-and-conquer
    by creating a “model”
    (which word need not and indeed generally
    ought not escape the lecturer’s mouth):
    *first* we’ll pick the president (the choices are 12),
    *second*, a vice (11)…

    the physical manifestation of our model,
    i now make it a point to remark, might
    very well look like

    _ _ _ _

    : four spots on the board
    signifying, first, hmmm.

    well, this is where it gets interesting
    because you’ve got to keep trying to
    pop people’s expectation-balloons…
    which *will* keep getting in the way…
    while still radiating confidence that
    once it all makes sense, it’ll be,
    and i make a point never to write this
    on the board without quote marks,
    “easy”.

    once you know even a *little*
    of what’s going on, you’ll know
    those _ _ _ _ spots are gonna have
    *numbers* in ’em… but!

    during the lecture, the teacher… “they”, them right there
    without the gender… is gonna point at the spots
    on the board and say things like
    “when we’re right here, we’ve
    already picked a president
    and a vice, and so…”
    (blah, blah, blah… writes “10”).

    and they’re gonna do well to do so;
    i’d do it myself and have many times.
    and maybe they’ll even write
    P V D G
    under the S.O.B.’s…
    i’ll have done something like this
    many times too
    or maybe the old
    1st 2nd 3rd 4th
    or… you might as well,
    there’s no telling *what*’s
    gonna get in their notebooks anyhow
    1 2 3 4
    .

    now, to the *unprepared* mind

    _ _ _ _
    1 2 3 4

    and

    12 11 10 9
    __ __ __ __

    look so much alike that the wonder
    is that anybody gets through any
    of this alive.

    because, while, believe it or not,
    *i* can satisfy pretty much any
    reasonably patient speaker of english
    that they’ve understood what’s going
    on in this problem, with very little fuss,
    and be right, there aren’t very *many* people
    who’d make this (elaborate) claim and most
    of those who would are dead wrong.

    and once you put those scribbles
    on the board *and* start talking about ’em,
    there’s this double discourse.
    and the level of formality is always
    *changing* and is very often
    *contested*.

    nobody wants to talk about any of this
    and for good reasons. we just want to
    talk about solving *math* problems
    for hecksake; we’re math *majors*.

    anyhow, 11880. get a grip man.

    as to P(12,4).
    when you (okay, now *you’re* the gender-free
    teacher! see how *you* like it!) work with
    students who are preparing to be
    *tested* on this kinda thing,
    and they’re *ready*, they’ll
    comfortably rattle off an answer
    like this (using the notations of
    the exam-prep materials;
    \null_{12} P_4 maybe).
    this will *not* suffice for
    all examiners, of course,
    since
    P(12,4) = 12!/[4!(12-4)!]
    may be about as far as
    certain quit-trying-to-explain-its
    will ever get in “simplifying”
    this result without a calculator
    (and somebody to punch its
    buttons for them; you will
    be blamed for this).

    2)
    forty-bang over product tenbang,tenbang,tenbang,tenbang.

    i’m very tired. this is harder than it looks.

    • i’m not sure what your point is, but #2 is 40 flags, 10 of each color, how many ways can they be hung in a line? if you’re talking examples.

      what i got out of this is – when we try to explain these things the students have to hold different levels of abstraction in their heads at once, and that’s part of what makes it hard to get, until you get it.

  8. right: the flags-on-a-pole problem is
    another version of this situation.
    there are
    10 Aqua(lung)
    10 Blue (\”oyster c\”ult)
    10 Creem-colored
    and
    10 Deep purple
    flags etcetera (AAA…DD).
    the “letters” version has the virtue…
    or misfeature… “one less layer
    of abstraction”: the objects we
    “move around” on the page
    actually *are* letters.
    (we might use colored pencils
    or chalk for eyecandy vividness
    for purposes of presentation
    and even make it our life work
    but we won’t be doing math but
    page design or animation or
    selling selling selling all the time).

  9. 40!/[10!10!10!10!]
    (shorter but harder to type;
    a comment on my earlier comment).

  10. 3.
    An “outfit” for the day consists of a choice of T-shirt, a pair of jeans, and a pair of shoes. I have 30 T-shirts, 4 pairs of jeans, and 2 pairs of shoes to choose from. How many different “outfits” are possible?

    30*4*2 of course:
    the “multiplication principle”
    at (almost) its simplest.

    in particular, the situation is simpler than
    that of problem 1. — where in some sense,
    the “choices” (but not the *number* or choices)
    associated with the various spots of the
    __ __ __
    diagram are *not* independent
    (a person chosen for president can’t
    be *also* chosen for vice-president,
    but the choice of T-shirt does *not*
    affect the choice of jeans.

    one introduces the matter of “independence”
    at one’s own peril here of course…
    my practice was to talk about it
    a little bit *without* “reading it
    into the notes” (by writing about
    it on the board). we want…
    very much… for the students to
    see this as a very simple situation.

    it’s more or less standard to present situations like this
    with so-called “tree diagrams”. this shouldn’t be overlooked
    altogether but isn’t the be-all-end-all that you’d be led
    to expect from some presentations: just as with the
    fill-in-blanks notation i was fussing about upthread,
    there are certain “hidden assumptions” about filling
    these diagrams in… and making ’em explicit isn’t
    *necessarily* the direction of “more clarity”.
    remember: students don’t like to read (and
    *particularly* don’t like to read *unfamiliar material*.

    4.
    Write down the sample space (16 outcomes) for the experiment “toss a coin four times”.
    a.
    What is the probability that exactly two heads occur?
    b.
    What is the probability that at least two heads occur?

    hhhh
    hhht
    hhth
    hhtt
    hthh
    htht
    htth
    httt
    thhh
    thht
    thth
    thtt
    tthh
    ttht
    ttth
    tttt

    a) #{hhtt, htht, htth, thht, thth, tthh} = 6
    (read ’em off the list!)
    so the probability is 6/16 (= 3/8)

    (full credit for .375 of course though
    this is to be discouraged… decimals
    don’t always “work out exactly”…)

    b) read ’em off the list again: 11/16.
    it may (or may not) be worth mentioning
    in a given context that one has
    {4\choose 0}+ {4\choose 1} + {4\choose2}
    in the numerator here.

    writing out the list itself?
    i’ve long lost track of the number
    of classes i’ve “drilled” this into.
    for the “just show me what to do” crowd:
    compute the number of “outcomes”: 2^4 =16.
    then write out (downwards) h, t, h, t, …
    until you’ve got 16 lines.
    go back to the top line and add (at the front
    if you want my exact display) a letter to each
    line; follow the pattern h, h, t, t, h, h, …
    (“counting by twos”). then count by 4’s; then 8’s.
    done.

  11. 5.
    (see the first comment in this long thread.)

    6.
    A “Texas Hold’em” hand is a set of two cards chosen from a standard deck. How many different Hold’em hands are there?

    How many Hold’em hands are “high pairs”—defined here as a pair of Jacks, Queens, Kings, or Aces?

    {52 \choose 2} = (52*51)/(2*1) = 1326

    4*{4\choose 2} = 24

    7.
    How many distinguishable arrangements are there for the letters of “ELEEMOSYNARY”? (Bonus Point: what does this word mean?)

    12!/(3!*2!) = 39 916 800

    “eleemosynary” means (something like) “charitable”.
    (nobody knew this on the day if i recall correctly.)

  12. 8.
    A billionaire awards large cash prizes to 5 lucky banjo players. On Monday, he will give away 1 million dollars; on Tuesday, 2 million; and so on until Friday, when he will give away 5 million. There are 11 banjo players on his short-list. In how many ways can the prizes be awarded?

    11*10*9*8*7
    (from first principles) or
    P(11, 5)
    (using the “permutation formula”).

    the dollar amounts are of course there
    to make “order matters” vivid.
    everybody’s always thinking about money.




Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s



%d bloggers like this: