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Unformatted text preview: Counﬁng Trees — tool used in a twostage or multi—stage experiment. Useful as long as the
tree is not too large. Totat number of terminating branches gives the totai
number of possibie outcomes. Example: What is the probability of getting at least two successive heads when
a coin is tossed three times? H<H Sample Space: p = 1/8 for each
Z . T HHH THH
<“ is? an it;
T<H<¥ HTT , TTT
T :5] P(2 successrve heads) = 3/8 Fundamental Theorem of Counting In a k~stage experiment, ii there are in ways to do the first step, n2 ways
to do the second step, . . . nk ways to do the kth step, th_eg there are
(n1)(n2) . . . (nk) possible outcomes. (Multiply the different number of ways.) Examgle: How many 3 digit numbers can be formed from the set of odd digits
given that the digits may be repeated; i.e., 555 is possible? There are 5 different odd digits giving 5 possibilities for the hundreds position, 5
pessibilities for the tens position, 5 possibilities for the ones position. Therefore,
there are (5)(5)(5) = 125 different numbers. Same question, but the digits cannot be repeated?
(5)(4)(3) = 60 different numbers. Permutation  a count of the number of ordered arrangements or sequences of
n distinct objects taken r at a time; i.e., cab is different from abc. Permutations  n!
occur when there is no replacement of the element used: ”P; e ( )l
h! — 7' . If all items are to be used (n = r), then the permutation becomes HP, = n! Example: In how many ways can a picture of 4 people be made if there are 6
people available? 1 i
531: 6' 96—2360
(6—4)! 21 Example: In how many ways can a set consisting of 5 different books on
mathematics, 3 on physics, and 2 on chemistry be arranged on a straight shelf
(that has space for these 10 books) if the books on each subject are to be kept
together? if the books are not to be kept together by subjects? 5! arrangements of mathematics books; 3! arrangements of physics books; 2!
arrangements of chemistry books; 3! arrangements of three types of books, so,
3![(5i)(3!)(2l)] = 8640 arrangements. If the subjects are not kept together, ali10 books are on a single shelf:
10! 10! 10! 1039 = ——m ~ #— = 3,628,800 arrangements.
(10—10): 0! 1 Combination — a count of the number of groupings containing r objects taken
from n available objects. Order of appearance of the objects does not create a
different grouping; i.e., cab is treated the same as abc, mainly each is the set {a,
b, c}. There are no repeats; once an object has been used, it cannot be reused: n! _ ”Pr «  n n r r!(n~—r)! *‘T‘
t‘, Example: How many different subcommittees of 5 can be formed from a club
with 16 members? i l
n=16andr=5t ,6C5m————w~16' =—1§'—'=4368
$0645)! Sill! Example: From 7 men and 4 women serving on a committee, how many
different subcommittees can be selected consisting of (a) 3 men and 2 women,
(b) 6 people with at ieast 3 men, (c) 6 people with at most 2 women. (a) Find the number of combinations of 3 taken from 7 for the men and the
number of combinations of 2 taken from 4 for the women. Multiply these
together (fundamental theorem of counting) to find the number of subcommittees made up of exactly 3 men and 2 women: (7C3)(4C2) = 210 subcommittees. (b) To find the number of subcommittees with at ieast 3 men out of 6 people,
add together the counts of exactly 3 men and 3 women, of exactly 4 men and 2
women, of exactiy 5 men and 1 woman, and of exactly 6 men and no women: (7C3)(4C3) +(,C,,)(,C2) +(7C5)(4C1) +(7C6)(4C,,) = 441 subcommittees. (c) To find the number of subcommittees with at most 2 women out of 6 people,
add together the counts of exactly 2 women and 4 men, of exactly 1 woman and
5 men, and of exactly no women and 6 men (see above): (4C2)(,C,,) +(4C1)(7C5) +(4C0)(,C6) = 301 subcommittees. Ni Permutation when some obiects are indistinguishable: ﬂ
711.142....nk. Example: How many arrangements of the letters in MISSISSIPPI are possible? ' N = 11 letters; n1 = 1 letter M; n2 = 4 letters 1; ns = 4 letters 8; m = 2 letters P, so I
—11—'—— = 34,650 ll4l4l2! If there are two indistinguishable objects (like heads or tails, red or white,
boy or girl) then the permutation is the same as the combination. The formula I
becomes —N—'— ,the formula for the count of the number of permutations when "1112‘
objects are indistinguishable. Also, since n1 + n2 = N and n2 = N  m, the formula
I
becomes 37%, the combination of N things taken n1 at a time MON.
"1 Counting and Probability Example: Toss a fair coin ﬁve times. What is the probability that (a) all heads or
all tails will appear? (b) exactly three heads will appear? (a)%(.5) +%(5) =1l16 l
(b) N = 5; ml: 3 which means nr"  2 and 3—2— = 10' Is the number of arrangements of three heads and two tails.5 The probability of one arrangement
HHHTT for example is (112)3(1/2)2 or (112)? So the probability of exactly three
heads' IS 10(1/2)5 — 10/32. (This iS the Binomial which will be seen later.) Example: What is the probability that a bridge hand of 13 cards will contain all
four aces and all four kings? The combination of n = 52 items taken r = 13 at a time gives ali possible
combinations for bridge hands, 52013, the denominator. The combination of n = 4
items taken r— = 4 at a time counts the ways to draw all four aces and all four
kings respectively. There are 44 cards remaining after subtracting the aces and
kings so the remaining cards to make up the hand of 13 cards come from these, 44C13 The probability is: W = 1.71 x 105, very unlikely so don't hold
52 13 your breath! (This is the hypergeometric which will be seen later.) Problems 1. You are studying for an exam by studying a set of 12 questions. You know 8
of these questions. For the exam, the instructor will select 5 questions at
random from the listof 12. If each question is worth 20 points, what is your
probability of making 80% or better? {.4242} 2. According to a theory of genetics, a certain cross of guinea pigs will result in
red, black, and white offspring in the ratio of 8:4:4. Find the probability that
among 8 such offspring, 5 wiil be red, 2 black, and 1 white. {21/256} 3. How many bridge hands are possible containing 4 spades, 6 diamonds, 1
club and 2 hearts? {1,244,117,160} 4. At a pizza shop 60% of the customers eat in the restaurant and 40% use the
carry out service. Suppose that 5 independent customers enter the pizza shop
in sequence. (a) What is the probability that the fifth customer is the first one who
eats in the restaurant? {.01536} (b) What is the probability that at least two of
the five customers use carry out service? {.66304} (c) if there are two
waitresses, each making $250.00 per week plus tips and there are 300
customers in an average week with those who eat in giving an average tip of
$4.00. What is the average yearly income (use 50 weeks) for each waitress?
{$30,500} 5. A labor dispute has arisen concerning the alleged unequai distribution of 20
laborers to four different construction jobs. The first job (considered to be
abominable employment) required 6 laborers; the second, third and fourth
utilized 4, 5, and 5 respectively. The dispute arose over an alleged random
distribution of the laborers to the jobs that placed all 4 members of a particular
ethnic group to job 1. In considering whether the assignment represented
injustice, a mediation panel desired the probability of the observed event. Find '
the probability of the observed event if it is assumed that the laborers are
randomly assigned to jobs. {1/323} 6. A manufacturer uses 3 trucking firms, A, B and C. From years of experience
the foiiowing probabilities have been assigned: Trucking Firm Proportion of the deliveries made Probability of a late delivery A 0.5 0.05
B 0.3 0.02
C 0.2 0.10 (a) if 10 shipments go out on a given day, what is the probability that 5 are
shipped by A, 3 by B and 2 by C? {.08505}  (b) if a customer tells the manufacturer that a shipment was delivered late, what
is the probability that firm C made the shipment? {.392} 7. Records show that 3 out of 10 people recover from a certain disease. Of 6
people who have this disease, what is the probability that (a) exactly 4 will
recover? (b) at least 2 will recover? {.0595, .5798} 8. A class in advanced physics is comprised of 10 juniors, 30 seniors, and 10
graduate students. The final grades showed that 3 of the juniors, 10 of the _
seniors, and 5 of the graduate students received A for the course. If a student is
chosen at random from this class and is found to have earned an A, what is the
probability that he or she is a senior? {5/9} 9. Given the set of numbers, {2, 3, 4, 5, 6}, find the following: (a) What is the
probability that the sum of two numbers picked at random from this set is an odd
number? (b) What is the probability that the sum of two different numbers
picked at random is an even number? {12l25, 215} 10. You are sick one night and awake to take some aspirin. You go to the
bathroom and pick one of 3 bottles and take a pill (one bottle contains poison,
two contain aspirin). You get sicker and show symptoms that 80% of the people
have after taking poison and 5% of the people have after taking aspirin. What is
the probability that you took poison? {8/9} 11. in a group of 50 applicants (for 10 different positions), there were 20 females
and 30 males. Of the 10 selected for the position, there was only one female.
Another female applicant is considering filing a discrimination suit against the
company. Based on probability and assuming all 50 applicants are equally
qualified for the positions, is the suit well founded? One way to respond to this
question is to answer the following: Is the probability that at most one female
would have been chosen less than .05? {Yes, the probability is .0308 so that a
legal suit is possible} 12. A new test has been designed to test sensitivity to chloral hydrate. it is
known that 62 people out of 1,000 become hyperactive when this sedative is
used. Of those individuals who are sensitive to chloral hydrate, the new test
confirms this sensitivity with a probability of .99; of those individuals shown not to
be sensitive to chloral hydrate, the new test shows no allergic reaction with
probability .999. How good is this new test? {Given that a person selected at
random reacts positively, the probability is .985 that he/she is in reality sensitive
to chloral hydrate so the test is pretty good} 13. What is the probability that a bridge hand of 13 cards will contain the ace,
king, queen, and jack of one of the suits? {.010564} 14. Urn one contains 12 red balls and 8 green balls. Urn two contains 8 red
balls and 10 green balls. One ball is taken from urn one and put in urn two,
Then a ball is taken from Urn two and put in Urn one. You select a ball from Urn
two. What is the probability that this ball is red? {.4526} 15. ln a group of 5000 people attending a conference, 40 % are Democrats,
40% are Republicans, and 20% are not committed to a particular party. 0f the
Democrats, 30% are women; of the Republicans, 40% are women; of those not
committed 25% are women. (a) How many men and how many women are
attending the conference? (b) The door prize at the opening assembly is
awarded randomly. What is the probability that the female who won the door
prize is Republican? {1650 women, 3350 men; .485} 16. How big is "30!"? is 30f bigger or smaller than Avogadro's number? How
many drops of water are there in the oceans of the world? How many grains of
sand are there on the beaches of the world? What information do you need to
answer these questions and where do you get that information? How do these
four numbers compare? 17. Find an algorithm for determining the number of zeros at the end of N!
where N is any whole number. Clearly give the steps in your algorithm. 18. How many paiindromes are seen on a digital watch during any 12 hour
period? A palindrome is a number that is read the same both forward and
backwards, like 101. {57‘} 19. Why is there no year 0? 20. Roll a tetrahedron die with numbers 1, 2, 3, 4. (a) What is the probability that
you will get exactly four "3"s in seven tosses? (b) What is the probability that you
will get at most two "3"s in seven tosses? (c) Alice and Ted roll the tetrahedron
until a 4 appears. Alice rolis first, then Ted. What is the probability that Alice will
win? This game could go on indeﬁnitely. {.057678, .7564, 4/7} 21. How long would it take to do the number of arrangements necessary for 15
people to be in a picture (sitting in a row) if one arrangement could be done
every 15 seconds? Compare this number to something in everyday life.
{623,700 years} 22. A fair coin is tossed 6 times. (a) How many ways can there be exactly four
heads and two tails? (b) How many ways can there be at most three tails? (c)
What is the probability that there will be the same number of heads as tails? {15,
42, 5/16} ...
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 Spring '11
 Bailey
 Counting

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