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Study Guide for Mathematics for Business Decisions I
This guide provides a summary of terms, formulas and symbols, as well as practice exercises
that the student may find helpful in preparing for the final examination in Math115a. The
student should keep in mind that this guide is NOT exhaustive. It is strongly recommended
that the student consult their class notes, the electronic text and any other materials that may
have been discussed or distributed by their instructor during the semester.
Key Terms
Complementary event
Conditional probability
Dependent events
Events
Experiment
Independent events
Intersection
Mutually exclusive events
Partition
Sample space
Tree diagram
Union
Venn diagram
Binomial / Bernoulli experiment
Binomial random variable
Continuous compounding
Continuous random variable
Continuous probability distribution
Cumulative probability function
Discrete compounding
Discrete random variable
Expected value
Exponential probability distribution
Exponential random variable
Future value
Histogram
Mean
Present value
Probability distribution
Probability distribution function
Probability mass function
Probability density function
Random number generator
Random sample
Range
Relative frequency
Representative sample
Uniform probability distribution
Key Formulas (Selected formulas will be provided on the exam.)
 
P A  P AC  1
P A  B  P A  PB  P A  B
P A  B  0
P A  B  P A  PB
P A  B 
P A | B  
P B 
P A  B  P APB | A  PBP A | B
P A | B  P A
P A  B  P APB
Complementary events
Additive Rule
Mutually exclusive events
Additive Rule for mutually exclusive events
Conditional probability
Multiplicative rule
Independent events
Multiplicative rule for independent events
n
a
k 1
Summation notation
k
1 n
 xk
n k 1
n
x
k 1
k
Sample mean
 f X xk 
Expected value of X
1
 u if 0  x  u

f X x   
 0 otherwise


Uniform density function
0
x
FX  x   
u
1
Uniform cumulative distribution function
if x  0
if 0  x  u
if u  x
 1  x /  if x  0
 e
f X x   

 0 otherwise
Exponential density function
 0 if x  0
FX x   
x / a
if x  0
1  e
Exponential cumulative function
Symbols
S
P A
A B
A B
AC
Sample space
Probability of A
x
Sample mean
Mean of exponential random variable
Probability mass / density function

f X x 
FX x 
A intersect B (both A AND B occur)
A union B (either A or B or both occur)
A complement (the event that A does not occur)
Cumulative distribution function
Practice Exercises
1.
You figure you have a 70% chance of getting an “A” in math, a 60% chance of getting an
“A” in rocket science and a 10% you won’t get an “A” in either class.
a. Construct the Venn diagram for this example
b. Find the probability of getting an A only in Math using the diagram.
c. Find the probability of getting an A only in Math using algebra and the formulas taught
in class.
2.
On a blind date, there is a 30% chance that you’ll like your date, an 80% chance that
you’ll like the food and a 95% chance that you’ll like one or the other. Find the probability
of liking your date but not the food.
3.
A department store has a box that contains a total of 100 balls. Some are colored green,
blue and red. As a sales promotion, a customer enters the store and selects a ball at
random from the box. If the ball is green, the customer is given $10 off a purchase of
$100 or more; if the ball is blue, the customer is given $25 off, and if the ball is red, the
customer is given $50 off a purchase of $100 or more. The ball is then returned to the
box.
a. In the box are 70 green, 20 blue and 10 red balls. How much can one expect to be
discounted if they select a ball?
b. Keep the total number of balls at 100 but change the proportion of colored balls so
that one can expect to be discounted $15 for picking out a ball. (Note that there are
many possibly answers. Try finding an appropriate formula, or try educated trial and
error.)
4.
On average, the probability that a student selected at random will get an “A” in
mathematics is .15, the probability that he’ll get an “A” in physics is .12, and the
probability that he’ll get an “A” in both classes is .08.
a. If the student gets and “A” in physics, find the probability that he’ll get an “A” in math.
b. If the student gets and “A” in math, find the probability that he’ll get an “A” in physics.
5.
Acadia Bank is considering whether to enter into a workout arrangement or foreclose on a
commercial loan with John Sanders who is behind in his payment. The following data
shows the past records of loan work outs (number of borrowers) where:
S and F are the events that the work out succeeds and fails, respectively,
Y, T and C are the events (or the properties) that the borrower has the 7 years experience,
that he has the bachelor's degree, and that the economic condition is normal, respectively.
For example, BR Bank has records of 3249 work outs for the clients in which 1470 were
successful and 1779 were failure. And among these 239 clients have 7 years experience in
which 105 were successful and 134 were failure.
Let X, ZY and Z be the amounts of money that the bank receives from a loan work out to a
randomly selected borrower, to a borrower having property Y (that is, with 7 years
experience), and to John Sanders who has all three properties (Y, T and C), respectively.
Answer the following based on the records. Assume that the full value is $4,000,000, the
default value is $250,000, and the foreclosure value is $2,100,000.
BR Bank (Y: 7 years)
Cajun Bank (T: bachelor)
Du Pont Bank (C: normal)
Total
S
F
1470 (105)
962 (510)
1386 (807)
3818
1779 (134)
1212 (644)
1417 (740)
4408
Total
3249 (239)
2174 (1154)
2803 (1547)
8226
a.
Estimate P(S ) and P (F ) , then compute E ( X ) . State your conclusion (work out or
foreclose) based on this expected value.
b.
Estimate P ( S | Y ) and P ( F | Y ) , then compute EZY  . State your conclusion based
on this expected value.
c.
Estimate P (Y | S ) , P (T | S ) and P(C | S ) . Note only BR Bank has data for clients'
experience; other two banks do not. Similar remarks apply for Cajun Bank (clients'
education) and DuPont Bank (state of economy).
d.
Assuming that Y, T and C are independent given S, compute PY  T  C | S  .
e.
Similarly for (c) and (d), compute PY  T  C | F  .
f.
Compute PS | Y  T  C  and PF | Y  T  C  using Bayes' Theorem and the
probabilities you found in (a), (d) and (e).
g.
Based on the probabilities you found in (f), compute E (Z ) . State your conclusion
based on this expected value.
Now, let's compute PS | Y  T  C  and PF | Y  T  C  using a different method.
According to the definition of the conditional probability
P( S | Y  T  C ) 
P( S  Y  T  C )
P(Y  T  C )
h.
Compute PS  Y  T  C  using (a) and (d). (In fact, this is already done in (f).)
i.
Estimate P(Y ) , P(T ) and P (C ) .
j.
Assuming that Y, T and C are independent, compute PY  T  C  .
k.
Compute PS | Y  T  C  using (h), (j) and (Cond).
l.
Similarly for (h)-(k), compute PF | Y  T  C  .
m. Based on the probabilities you found in (k) and (l), recompute E(Z). State your
conclusion based on this expected value.
6.
Evaluate:
100
99
k 1
k 1
 (2k  1)   (2k  1)
7.
H.F. Credit Union offers a “high yield” savings account with effective annual yield 1.41%.
BOA Bank offers a Money Market savings account with an annual interest rate of 1.44%,
compounded monthly. Which account has a higher yield?
8.
Disney stock closed at $17.02 yesterday (3/31/03). If this is a 2.2% decrease from last
week (3/24/03), find last week’s closing price.
9.
You estimate that a 24-week option will be worth $6 per share on its expiration date. If
the current annual risk-free rate is 5%, find the option’s value on its start date. (Round to
the nearest cent).
10. The histogram below represents the years in business data from the BR Bank clients (from
project 1). Note that the relative frequencies listed above the bars have been rounded, so
they are approximate.
0.15
0.14
0.13
0.12
0.11
0.08
0.06
19-20
15-16
13-14
11-12
9-10
7-8
5-6
0.06
17-18
0.08
0.07
3-4
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
1-2
relative frequency
Years In Business
years
a. What is meant by “relative frequency” along the vertical axis?
b. If there were 3249 borrowers represented in this data, approximately how many
borrowers had 19 – 20 years in business?
11. Determine if the random variables are finite or continuous:
a. X records the length of time in seconds and fractions of seconds between consecutive
vehicles that pass by a particular intersection ______________________.
b. X records the number of hours in a 3 month period in which an individual works at her
part time job __________________.
c. R records the sum of the money from a box of coins __________________.
12. Let A be the age of a randomly selected student in your English class. You find the ages of
seven students sitting in your row are 20, 21, 25, 19, 18, 21, and 20 years.
a. Use this information to estimate E ( A) .
b. Find FA (20) .
c. Give a practical explanation of the value of FA (20) in terms of this specific problem.
13. Y is an exponential random variable that records the waiting time in minutes between
consecutive customer arrivals at a particular store. The expected value of Y is 12 minutes.
Find the following:
a. Find the probability that the time between consecutive arrivals of customers is more
than 10 minutes.
b. Find the probability that the time between consecutive arrivals of customers is exactly
13 minutes.
c.
14.
fY  4  _____________
d.
FY  4  _____________
Let X be a random variable that records the number of heads in 4 tosses of a coin.
a. What is the sample space for this experiment?
b. Is this a discrete random variable or a continuous random variable? Explain.
c. Is this an example of a Binomial Random Variable? Explain.
d. Use a graph to illustrate f X  x  and FX ( x) .
e. f X  3 = _____________
15.
d.
FX (3)  _____________
Suppose that X is exponentially distributed with a mean 1/2. What is P(X>1)?
16.
For each of the following, decide which could be a p.m.f. (Probability mass function), a
p.d.f. (Probability density function), a c.d.f. (Cumulative density function) in this indicate
if it is for a discrete or a continuous random variable, or none of these. If the answer is
none, explain why.
a.
b.
0.6
0.5
0.4
0.3
Series1
0.2
0.1
0
1
2
3
4
c.
5
6
d.
x
0
1
2
3
4
f X  x
0.1
0.1
0.3
0.4
0.1
e.
f.
17. A supplier of kerosence has a 150 gallon tank that is filled at the beginning of each week.
His weekly demand shows a relative frequency behavior that increases steadily up to 100
gallons and then levels off between 100 and 150 gallons. If Y denotes weekly demand in
hundreds of gallons, the pdf is given by:
 y 0  y 1

fY  y   1 1  y  1.5
0 elsewhere

a. Verify that fY  y  is a pdf.
b. Determine the probability that the weekly demand is greater than 1.
c. Determine the probability that the weekly demand is less than 1.
18. A baseball player is a 0.300 hitter. He comes to bat three times in a game.
a. How many hits do we expect the player to get?
b. What is the probability that he will get exactly one hit?
19. The probability that a patient recovers from a delicate heart operation is 0.9. There are 4
patients in ICU recovering from their heart surgery. Let X record the number of patients
that survive.
a. Give the probability mass function for X.
b. Give the cumulative density function for X.
c. What is the probability that at most two survive?
d.
f X  4  ____________
e.
FX  3 
20. Suppose X is a continuous random variable and the density curve is given in graphical
form. Shade in the region under the following density curves that correspond to the
following probabilities:
a.
b.
P  X  2
P 1  X  2
c.
d.
P 1  X  4
P  X  4
21. You were told that a car broke down between 2:00 pm and 5:00 pm and that any time
within this interval is likely to be the exact failure time of the car. Thus you would use the
following probability density function,
1 2  x  5
f X  x   3
0 otherwise
where x is the exact time the car failed between 2:00 and 5:00 pm.
a. What is the probability that a car failed after 3:00?
b. What is the probability that a car failed at exactly 2:30?
c. What is the probability that a car failed between 2:20 and 4:00?
22. Suppose that a certain mathematics class has 28 students. Of these, 14 are first-year
students, 17 are business major, and 8 are neither.
a. Suppose that a business student is selected at random. What is the probability that he
or she is also a first year student?
b. Suppose that student from this class is selected at random. Given that he or she is not
a first-year student, what is the probability that he or she is business major?
23. Thirty percent of employees of some company are college graduates. Of all its employees,
25% earn more than $35000 per year, and 18% are college graduate earning more than
$35000 per year.
a. What is the probability that randomly selected employee earns more than $35000 per
year, given that he or she is college graduate?
b. What is the probability that the randomly selected employee is college graduate, given
that he or she earns less or equal to $35000 per year?
24. Of the students at certain college, 50% regularly attend the football games, 30% are
freshmen, and 40 % are upperclassmen who do not regularly attend football games.
a. Suppose a student is selected at random. What is the probability that the person is
both a freshman and regularly attends football games?
b. Suppose a student is selected at random. What is the probability that the person is a
freshman, given that he or she regularly attends football games?
c. Suppose a student is selected at random. What is the probability that the person is a
freshman, given that he or she does not regularly attends football games?
25. Let E be the event that an applicant for a home mortgage is employed, C be the event that
she or he has a good credit rating. We know that 97% of the applicants with good credit
are employed. The probability that an applicant has a good credit is 0.7. The probability
that an applicant is employed or has a good credit is 0.95.
a. Find the probability that an applicant for a home mortgage is employed.
b. Find the probability that an applicant who is employed also has a good credit rating.
26. Suppose the probability that a person will become unemployed the next year is 0.09. The
probability that a person will have high school diploma is 0.95. The probability that a
person either will become unemployed the next year or will have high school diploma is
0.98.
a. Find the probability that a person will become unemployed the next year but will
have high school diploma. (Include any notation that is necessary)
b. Given that a person will become unemployed the next year, what is the
probability that he or she will have high school diploma?
c. What percent of people with high school diploma will be unemployed?
27. Suppose the probability that a business student will pass Math 115A is 0.93. The
probability that a business student will pass Accounting Class is 0.92. The probability that
a business student either will pass Math 115A or will pass Accounting Class (or both) is
0.95.
a. Find the probability that a business student will pass both classes. (Include any
notation that is necessary)
b. What percent of the students who will pass Accounting Class will also pass Math
115A? (Include any notation that is necessary)
28. A printer has three bookbinding machines. The table below gives the proportion of the
total book production for each machine and the probability that the machine produces a
defective binding. Suppose that a book is selected at random and found to have a
defective binding. What is the probability that it was bound by machine 2?
Machine
Proportion of
books bound
Probability of
defective binding
1
0.4
0.03
2
0.1
0.06
3
0.5
0.01
29. A mortgage company rated 75% of its applicants as having good credit and 25% as having
bad credit. Of the good credit ones, 95% will pay the loan back. They estimated that only
3% of those with bad credit will pay the loan back. Let G be the event that an applicant
has a good credit rating and S be the event that an applicant will pay the loan back. What
percent of all applicants who will pay the loan back have good credit?
30. A researcher surveys patients who use a hospital emergency room. Suppose I is the event
that a randomly selected patient has medical insurance and that H is the event that a
randomly selected patient requires hospitalization. The researcher estimates that 30% of
patients in emergency room require hospitalization. The researcher estimates that 60% of
those patients in emergency room who require hospitalization have medical insurance. Of
those who do not require hospitalization 65% have medical insurance. What percent of
those patients in emergency room who have insurance require hospitalization?
31. You are trying to develop a strategy for investing in two different stocks. The anticipated
annual return for a $1,000 investment in each stock has the following probability
distribution:
Probability
0.1
0.3
0.3
0.3
RETURNS
Stock A
-$100
0
$80
$150
Do you think you will invest in Stock A or Stock B? Explain.
Stock B
$50
$150
-$20
-$100
32. The game of roulette is played with a wheel containing 37 slots. The slots are numbered
with the integers from 0 to 36 inclusive. A player may place a $1 bet (or any other
amount) on any number. If the ball falls in the slot bearing that number, the player
recovers his bet and receives in addition 35 times the amount wagered, otherwise, he
losses his bet. Using a $1 bet, find the expectation for this game.
33. The CNA Insurance Company charges Mike $250 for a one-year $100,000 life insurance
policy. Because Mike is a 21-year-old male, there is a 0.9985 probability that he will live
for a year (based on the U.S. National Center for Health Statistics).
a. If Mike purchases the policy what is his expected value?
b. What would be the cost of the insurance policy if the company just breaks even (in the
long run with many such policies), instead of making a profit?
34. Consider the numbers game starting many years ago by organized crime groups and now
run legally by many organized governments, as well as by some governments that aren’t
so well organized. Often called a “Pick Three” game, you place a bet that the three-digit
number of your choice will be the winning number selected. The typical winning payoff is
499 to 1, meaning that for each winning dollar bet, you would be given $500; your net
return is therefore $499. Suppose that you bet $1 on the number 327. What is the
expected value of gain or loss?
35. A recent survey of investors with Internet access divided them into two groups, those who
trade online and those who do not (traditional investor), and found distinct differences
between them. Forty-eight percent of the traditional investors were bullish on the market,
that is, they believe that the market will trend higher, while 69% of the online investors
were bullish on the market. Suppose that the survey was based on 500 traditional
investors and 500 online investors.
a. Set up a 2  2 contingency table to evaluate the probabilities
b. Give an example of a simple event.
c. Give an example of a conditional event.
d. What is the complement of “being bullish on the market”?
What is the probability that an investor chosen at random
e. Is bullish on the market?
f.
Is an online investor?
g. Is bullish on the market and is an online investor?
h. Is bullish on the market and is a traditional investor?
i.
Is bullish on the market or is an online investor?
j.
Is not bullish on the market or is an online investor?
k. Is not bullish on the market and is a traditional investor?
36. Couples with 3 offspring were selected for a national survey. What is the probability that a
couple selected at random from this group has at least 2 girls. Assume that the probability
of having a boy is equal to the probability of having a girl.
37. The table below illustrates the mortality statistics of the infamous voyage of the Titanic.
Assume that one person is selected at random from the 2223 people aboard the Titanic.
Titanic Mortality
Men
Survived
332
Died
1360
Total
1692
Women
318
104
422
Boys
29
35
64
Girls
27
18
45
a. Find the probability of selecting a man or a boy.
b. Find the probability of selecting a man or someone who survived.
Total
706
1517
2223
38. In 1936, the Literary Digest magazine published a poll where it was predicted that Alf
Landon would receive 54% of the votes in the up coming presidential election and FDR
would receive 41%. In the election, FDR received 61% of the votes. The method used by
the Literary Digest was to, among other things, mass mail 10 million mock ballots in which
2.4 million were returned. Let R be the event that a voter is a Roosevelt supporter and let
A be the event that a voter answered the poll.


a. Find and explain P R | AC .
b. Find and explain P A | R.
39. What is the future value of $600 invested at 5% compounded daily after a period of 3
years?
40. How long does it take for an investment to double in value if it is invested at 8%
compounded continuously?
41. Which investment will be worth more: $1,000 invested at 5.9% compounded monthly or
$1,000 invested at 5.6% compounded continuously?
42. Steve wants to buy a used car for $5,000 in three years when he turns 16. How much
does he need to invest now if his investment can earn 5% compounded continuously?
43. An investment of $750,000 made in 1999 is worth $850,000 in 2002. What was the rate
of increase if compounded continuously?
4
 ( k +4)
44. Use properties of summation to find the sum:
2
K 1
45.
Write the series using summation notation:
2
3
4
12
+
+ ...+
3
4
5
13
10
46. A sum of a series is represented by:

5k +3. Write the sum of the same series using
k 4
summation notation where the index is k = 1.
47. You have the records of the monthly closing stock prices of your company for the last 8
years and want to create a histogram. You set up the bins and the frequencies. What
should be the sum of the frequencies?
48. Let X be a finite random variable with the following table of values for f x .
x
f x (x)
a. Plot the graph of the p.m.f. below.
1
2
3
4
0.1 0.4 0.3 0.2
b. Plot the graph of the c.d.f. below.
49. Suppose you are waiting for your roommate to come home from a date on Friday night.
Let X be the length of time after 9 PM that he/she will arrive home. He/she never stays
out after 2 AM. Assume that X has a uniform distribution.
a. Find a formula for FX (x) and use it to answer the following questions:
(i)
What is the probability that he/she arrives home between midnight and 1 AM.
(ii)
What is the probability that he/she arrives home before midnight?
b. Find a formula for f X (x) and use it to answer the following questions.
(i)
What is the probability that he/she arrives home between midnight and 1 AM.
(ii)
What is the probability that he/she arrives home before midnight?
c. Discuss the similarities and differences between the answers you gave to a and b.
50. Suppose that customers arrive at the checkout counter at a grocery store. Let X be the
continuous random variable that gives the time, in minutes and parts of minutes, between
the arrival of consecutive customers. It can be shown that this is an exponential random
variable and that the average time between customers is 45 seconds.
a. Find the probability that the time between the arrival of consecutive shoppers is
between 1 and 2 minutes.
b. Find the probability that the time between the arrival of consecutive shoppers is at
least 1 minute.
c. Find the probability that the time between the arrival of consecutive shoppers is at
most 1 minute and 40 seconds.
d. Find FX (30) and explain what it means in terms of how long it is between customers.
51. The graph below shows the number of flat tires the buyers of Lakerock Tires get in a year.
a. What kind of a graph is this?
b. What percent of the people get at least 3 flats during a year?
c. What percent of the people get less than 3 flats in a year?
52. Let T be a uniform continuous random variable on the interval [0, 5}.
a. Find a formula for the p.d.f., f T of T.
b. The p.d.f., f T has f T (3  T  5)  ________ .
c.
P(T  3)  ________.
d. The c.d.f., FT has FT (3)  ____________ .
53. A few of the historical adjusted weekly closing prices of a stock are shown below.
4/08/02
4/01/02
3/25/02
3/18/02
3/11/02
$86.89
$85.95
$84.28
$86.00
$84.93
a. Fill in the column with the ratios of adjusted closing prices.
b. Compute the mean of the given ratios.
c. Find the weekly ratio corresponding to a yearly risk free rate of 7%.
d. Create a column for the normalized weekly ratios.
54. An airport uses a moving sidewalk that is 600 feet long to assist passengers throughout
the terminal. Unfortunately it forgot to install handrails or side supports, so passengers
can fall off the sidewalk. When a passenger falls off, he or she is likely to do so at any
point along the sidewalk. Let X be the distance, in feet, from the start of the sidewalk to
the point where a passenger falls off.
a.
Assuming X is a continuous random variable, find the formulas for the probability
density function and cumulative distribution function. Include the graphs for both.
Label your graphs clearly.
b.
Suppose a passenger’s gate is 200 feet from the start of the sidewalk. Use the cdf to
determine the probability that the passenger will fall off before he or she reaches the
gate. Also illustrate this on the graph of the cdf.
c.
In the hopes of increasing sales, the owner of a coffee cart will move her cart to the
place where, on average, a passenger will fall off the sidewalk. Where should the cart
be positioned?
d.
Use the pdf to determine the percent of passengers that will fall off between the
coffee cart (positioned according to part C) and the restroom that is 500 feet from
the start of the sidewalk. Also illustrate your answer on the graph of the pdf above.
55. Suppose you have three dimes and five quarters in your left pocket while you have two
dimes and six quarters in your right pocket. You randomly select one coin from each
pocket.
a. Write out the sample space for this experiment.
b. Determine probabilities for each of the elements in your sample space.
c. If M is the event that you selected at least one quarter, find P(M).
d. Find E(T) if T represents the total amount of money you have taken out of your
pockets.
56. You plan to place a $4 bet on a horse race with only four horses named Slowpoke,
Wanderer, Deadlast, and Noshow. The type of bet you plan to make requires that you pick
the first place and second place horses in order.
a. Write out the sample space for all the possible bets that can be made of this type. Note
– this bet does not require that you pick the third and fourth place horses.
b. Suppose you collect $12 if Wanderer finishes in either first or second place and Noshow
finishes in either third or fourth place. Suppose you collect $6 if Slowpoke and Deadlast
finish in first or second place in either order. You will only lose your bet if anything else
happens. Let X represent your gain from a single bet. Find E(X)
57. After making calculations and running simulations, suppose we estimate the following
possible closing prices for our stock and the corresponding probability distribution. If the
strike price is $50, determine a fair price for the option by the method of project 2.
Closing price
Probability
$47 $49 $50 $54 $55
0.1 0.2 0.4 0.2 0.1
58. Assume we have the same scenario as in the pricing an option project (European call
option, etc.) After making calculations and running simulations, suppose we find only 3
possible closing values for the stock on the expiration date of the option, all of which are
equally likely. If the strike price for the option is $85 and the three possible closing values
for the stock are $82, $85, and $88 find the expected future value of the option.
Solutions to Practice Exercises
40. 600(1+.05/365)(365*3) = $697.09
42. 2=1*e(.08t)
ln2 = .08t
t=ln2/.08 = 8.66years
43. 1000(1+.059/12)12=$1,060.62
1000e.056= $1,057.60. The investment at 5.9%
compounded monthly will be worth more.
44. 5000e(-.05*3) = $4,305.54
45. 850,000=750,000e(r*3)
850,000/750,000=e(3r)
r=ln(850,000/750,000)/3 = .0417 or 4.17%
46.
4(4  1)( 2 * 4  1)
+ 4*4 = 30+16 = 46 or (12+4) + (22+4) + (32+4) + (42+4)
6
5 + 8 + 13 + 20 = 46
11
47.

k 1
k 1
k2
12
or

k 2
k
k 1
7
48.
ln(850,000/750,000=3r
 5k  18
k 1
49. Monthly records for 8 years: 12*8 = 96 records
=