Download p - Varsity Math by Coach G

BINOMIALPROBABILITYDISTRIBUTION
=np
2
 =np(1–p)
GEOMETRICPROBABILITYDISTRIBUTION
x=thenumberoftrialsuntilthefirstsuccessis
observed
p=probabilityof"success"onasingletrial
x 1
p ( x)  p 1  p  σ  np (1  p ) Mean(Expectedvalue)
Variance
2=
1 p
p2
=
1 p
p2
n
nk
p ( x  k )    p k 1  p  k
 
Mean(Expectedvalue)
Variance
StandardDeviation RANDOMVARIABLES
Mean=  xi  p  xi  StandardDeviation=   xi     p  xi  2
1.
Among the voters in a certain precinct of Dallas. 80% are
Democrats. If ten voters are selected at random, determine
the probability that:
a) exactly 5 are Democrats.
b) exactly 7 are Democrats.
c) at least 4 are Democrats.
d) what is the expected number of Democrats in a
sample of size 25?
e) what is the standard deviation for part (d)?
2.
The following data are based on information taken from
the Statistical Abstract of the United States. In this table, x
= size of family. The % data are the percentages of U.S.
families of this size.
x
2
3
4
5
6 7 or more
% 42% 23% 21% 10% 3%
1%
a) Convert the percentage data to probabilities and make
a histogram of the probability distribution for family
size.
b) What is the probability that a family selected at
random will have only two members?
c) What is the probability that a family selected at
random will have more than three members?
d) Compute , the expected family size (round families
of size 7 or more to size 7).
e) Compute , the standard deviation (round families of
size 7 or more to size 7).
3.
Approximately 71% of all college students claim that the
main reason they are attending college is to make more
money after graduation. In 1967, approximately 83% of
all college students claimed they were attending college
primarily to develop a meaningful philosophy of life. Let
x be the random variable which represents the first college
student selected at random who you encounter who says
he or she is in college primarily to make more money.
a) Write out the formula for the probability distribution
of the random variable.
b) Find the probability that x = 1, x = 2, x  3
StandardDeviation
=
1
p
4.
Approximately 3.6% of all (untreated) Jonathan apples
have a bitter pit – a disease of apples resulting in a soggy
core, which is caused either by over watering the apple
tree or a calcium deficiency in the soil. Let n be a random
variable that represents the first Jonathan apple chosen at
random that has bitter pit.
a) Find the probability that n = 3, n = 5. n = 12, and
n < 8.
b) What is the probability that it takes at least 5 apples
before you select one which has bitter pit?
5.
In a study of sleep patterns of adults, it has been found
that 23% of the sleep time is spent in the REM (rapid eye
movement) stage.
a) If a sleeping adult is observed 5 randomly selected
times, find the probability that exactly one of the five
observations will be made during REM sleep.
b) Find the standard deviation for the number of REM
stages observed in groups of five observations.
6.
a. Find the mean, variance, and standard deviation of X.
X
-1
0
1
2
P(X) 0.3 0.1 0.5 0.1
b. Find the mean, variance, and standard deviation of Y.
Y
2
3
5
P(Y) 0.6 0.3 0.1
c) Let W = 3 + 2 X. Find the mean, variance, and
standard deviation of W.
d) Let W = X + Y. Find the mean, variance, and
standard deviation of W.
e) Let W = X – Y. Find the mean, variance, and
standard deviation of W.
f) Let W = X + X. Find the mean, variance, and
standard deviation of W.
g) Let W = 2X. Find the mean, variance, and standard
deviation of W.
h) Let W = X – X. Find the mean, variance, and
standard deviation of W.
i) Let W = 2X + 5Y. Find the mean, variance, and
standard deviation of W.
7.
A stationery store has decided to accept a large shipment
of ball-point pens if an inspection of 20 randomly selected
pens yields no more than two defective pens.
a) Find the probability that this shipment is accepted if
5% of the total shipment is defective.
b) Find the probability that this shipment is not accepted
if 15% of the total shipment is defective.
8.
On the leeward side of the island of Oahu in the small
village of Nanakuli, about 80% of the residents are of
Hawaiian ancestry.
a) What is the probability that the first person you meet
in Nanakuli is of Hawaiian descent? The second
person? The third person?
b) What is the probability that you will meet at least 5
people before you meet someone who is of Hawaiian
ancestry?
9.
On Waikiki it is estimated that about 4% of the residents
are of Hawaiian ancestry. Repeat question #8 for Waikiki
residents.
10. At Fontaine Lake Camp on Lake Athabasca in northern
Canada, history shows that about 30% of the guests catch
lake trout over 20 pounds on 4-day fishing trip. Let n be a
random variable that represents the first trip to Fontaine
Lake camp on which a guest catches a lake trout over 20
pounds.
a) Write out the formula for the probability distribution
of the random variable n.
b) Find the probability that a guest catches a 20 pound
trout for the first time on trip number 3. On trip
number 2.
c) Find the probability that it takes more than 4 trips for
the guest to catch a 20 pound lake trout.
11. Police find that a patrol unit gets a 30% arrest record
when it sets up a checkpoint for drunk drivers.
a) Find the probability that of 200 drivers checked, there
will be exactly one arrest. More than 3 arrests?
b) What is the expected number of arrests? The standard
deviation of arrests?
c) What is the probability that the third driver is the first
one arrested? That at least 10 drivers are checked
before an arrest is made?
12. The Los Angeles Times (Dec. 13, 1992) reported that
what airline passengers like to do most on long flights is
rest or sleep; in a survey of 3697 passengers, almost 80%
did so. Suppose that for a particular route, the actual
percentage is exactly 80%, and consider randomly
selecting six passengers. Then x, the number among the
selected six who rested or slept, is a binomial random
variable with n = 6 and p = 0.8.
a) Calculate P(4)
b) Calculate P(6), the probability that all six selected
passengers rested or slept.
c) Determine P(x  4).
13. Refer to #12, and suppose that ten rather than six
passengers are selected (n = 10, p = 0.8).
a) What is P(8)?
b) Calculate P(x  7).
c) Calculate the probability that more than half of the
selected passengers rested or slept.
14. Twenty-five percent of the customers entering a grocery
store between 5 P.M. and 7 P.M. use an express checkout.
Consider five randomly selected customers, and let x
denote the number among the five who use the express
checkout.
a) What is P(2), that is, P(x = 2)?
b) What is P(x  1)
c) What is P(2  x)?
d) What is P(x  2)?
15. A breeder of show dogs is interested in the number of
female puppies in a litter. If a birth is equally likely to
result in a male or female puppy, give the probability
distribution of the variable
x = number of female puppies in a litter of size 5
16. Selected boxes of a breakfast cereal contain a prize.
Suppose that 5% of the boxes contain the prize and the
other 95% contain the message "Sorry, try again." A
consumer determined to find a prize decides to-continue
to buy boxes of cereal until a prize is found. Consider the
random variable x, where x = number of boxes purchased
until a prize is found.
a. What is the probability that at most 2 boxes must be
purchased?
b. What is the probability that exactly four boxes must be
purchased?
c. What is the probability that more than four boxes must
be purchased?
17. If the temperature in Florida falls below 32°F during
certain periods of the year, there is a chance that the citrus
crop will be damaged. Suppose that the probability is 0.1
that any given tree will show measurable damage when
the temperature falls to 30°F. If the temperature does drop
to 30°F, what is the expected number of trees showing
damage in orchards of 2000 trees? What is the standard
deviation of the number of trees that show damage?
18. Thirty percent of all automobiles undergoing an emission
inspection at a certain inspection station fail the
inspection.
a) Among 15 randomly selected cars, what is the
probability that at most 5 fail the inspection?
b) Among 15 randomly selected cars, what is the
probability that between 5 and 10 (inclusive) fail to
pass inspection?
c) Among 25 randomly selected cars, what is the mean
value of the number that pass inspection, and what is
the standard deviation of the number that pass
inspection?
19. You are to take a multiple-choice exam consisting of 100
questions with five possible responses to each. Suppose
that you have not studied and so must guess (select one of
the five answers in a completely random fashion) on each
question. Let x represent the number of correct responses
on the test.
a) What kind of probability distribution does x have?
b) What is your expected score on the exam?
c) Compute the variance and standard deviation of x.
20. Which of the following are continuous variables, and
which are discrete?
a) Number of traffic fatalities per year in the state of
Florida
b) Distance a golf ball travels after being hit with a
driver
c) Time required to drive from home to college on any
given day
d) Number of ships in Pearl Harbor on any given day
e) Your weight before breakfast each morning
f) Cost of an adult ticket at each of the movie theaters in
your town
21. Which of the following are continuous variables, and
which are discrete?
a) Amount of sleep you got last night
b) Home team score in a basketball game
c) Number of ducks sitting on a pond
d) BTUs absorbed by a solar panel
e) Volume of water in Lake Powell
f) Number of prisoners in the county jail
22. The National Hockey League keeps a day-to-day total of
all goals scored in the league. Games consist of three
periods. An overtime (OT) period is played in the event
that there is a tied score at the end of the third period.
Through March 20, 1990, there was a total of 5747 goals
scored in the NHL during the 1989-1990 season. The
breakdown by periods is as follows:
Period x
1
2
3
OT
Goals Scored f 1776 2035 1883 53
a) A scoring play is chosen at random from games
played in the 1989-1990 NHL season up through
March 20, 1990. Use relative frequencies to calculate
the probability P(x) that the goal was made in the x =
1st, 2nd. 3rd, OT period
b) Use a histogram to graph the probability distribution.
Note: Consider OT as a 4th period with x = 4.
c) Find the expected value of the distribution. How can
you interpret this value?
d) Find the standard deviation of the distribution.
e) In which period is a goal most likely to be scored? If
you picked a scoring play at random from the games
included in the table above, what is the probability
that the goal was not scored in the second period of a
game?
23. It is found that for one section of Interstate 30, 94% of the
vehicles are traveling at speeds greater than 65 mph. Find
the probability that among 24 randomly selected vehicles,
less than 20 are traveling above 65 mph.
24. The head nurse on the third floor of a community hospital
is interested in the number of nighttime room calls
requiring a nurse. For a random sample of 208 nights
(9:00 P.M. to 6:00 A.M.), the following information was
obtained, where x = number of room calls requiring a
nurse and f = frequency with which this many calls
occurred (i.e., number of nights).
x 36 37 38 39 40 41 42 43 44 45
f 6 10 11 20 26 32 34 28 25 16
a) If a night is chosen at random from these 208 nights,
use relative frequencies to find P(x) when x = 36, 37,
38, 39, 40, 41, 42, 43, 44, and 45.
b) Use a histogram to graph the probability distribution
of part a.
c) Assuming these 208 nights represent the population
of all nights at community hospital, what do you
estimate the probability is that, on a randomly
selected night, there will be from 39 to 43 (including
39 and 43) room calls requiring a nurse?
d) What do you estimate the probability is that there will
be from 36 to 40 (including 36 and 40) room calls
requiring a nurse?
e) Find the expected number of room calls requiring a
nurse.
f) Find the standard deviation of the x distribution.
25. The following data are based on information taken from
Daily Creel Summary (Feb. 28, 1993), published by the
Paiute Indian Nation, Pyramid Lake, Nevada. Movie stars
and U.S. presidents have fished Pyramid Lake. It is one of
the best places in the lower 48 states to catch trophy
cutthroat trout. In this table, x = number of fish caught in
a 6-hour period. The % data are the percentages of
fishermen who caught x fish in a 6-hour period while
fishing from shore.
x
0
1
2
3 4 or more
% 44% 36% 15% 4%
1%
a) Convert the percentages to probabilities and make a
histogram of the probability distribution.
b) Find the probability that a fisherman selected at
random fishing from shore catches one or more fish
in a 6-hour period.
c) Find the probability that a fisherman selected at
random fishing from shore catches two or more fish
in a 6-hour period.
d) Compute , the expected value of the number of fish
caught per fisherman in a 6-hour period (round 4 or
more to 4).
e) Compute , the standard deviation of the number of
fish caught per fisherman in a 6-hour period (round 4
or more to 4).
26. Almost all new independent businesses (not franchises)
that fail do so in the first 10 years. The following data are
based on information from the Statistical Abstract of the
United States (112th ed). For the population of new
businesses that fail, let x = the year in which the business
fails. For instance, x = 3 means the business fails in its
third year. The % data are the proportions of such
businesses that fail during the xth year of the business.
x
%
a) Make a histogram of the probability
1
0.02
distribution.
2
0.07
b) Find the probability that a business
3
0.15
selected at random that fails will fail
4
0.18
in its third year or before.
5
0.21
c) Find the probability that a business
6
0.16
selected at random that fails will fail
7
0.10
in its fifth year or later.
8
0.06
d) Compute , the expected value for
9
0.04
the year in which a business that
10 0.01
fails will fail,
e) Compute , the standard deviation
for the year in which a business that
fails will fail.
27. USA Today (June 2, 1993) reported that approximately
24% of all state prison inmates released on parole become
repeat offenders. Suppose the parole board is examining
five prisoners up for parole. Let x = number of prisoners
out of five on parole who become repeat offenders.
x
0
1
2
3
4
5
P(x) 0.254 0.400 0.253 0.080 0.012 0.001
a) Find the probability that one or more of the five
parolees will be repeat offenders. How does this
number relate to the probability that none of the
parolees will be repeat offenders?
b) Find the probability that two or more of the five
parolees will be repeat offenders.
c) Find the probability that four or more of the five
parolees will be repeat offenders.
d) Compute , the expected number of repeat offenders
out of five.
e) Compute , the standard deviation of the number of
repeat offenders out of five.
28. Mr. Dithers wants to insure his yacht for $80,000. The
Big Rock Insurance Company estimates a total loss may
occur with a probability of 0.005, a 50% loss with
probability of 0.01, and a 25% loss with probability 0.05.
If Big Rock will pay no benefits for any other partial loss,
what premium should Mr. Dithers pay each year if Big
Rock wants to make a profit of $250?
29. The student senate is sponsoring a car raffle to buy
playground equipment for disadvantaged children. The
senate buys a used car for $3000 and sells 3750 raffle
tickets at $1.50 per ticket.
a) If you buy 30 tickets, what is the probability that you
will win the car? What is the probability that you will
not win the car?
b) Your expected earnings can be found by multiplying
the value of the car by the probability that you will
win it. What are your expected earnings? Is it more or
less than the amount you paid for 30 tickets? How
much did you effectively contribute to buy
playground equipment?
30. Sophie is a dog who loves to play catch. Unfortunately,
she isn't very good, and the probability that she catches a
ball is only 0.1. Let x = number of tosses required until
Sophie catches a ball.
a) Does x have a binomial or a geometric distribution?
b) What is the probability that it will take exactly two
tosses for Sophie to catch a ball?
c) What is the probability that more than three tosses
will be required?
31. In a study of middle-aged adults (40 to 65 years), it is
found that 7.8% suffer from hypertension. A follow-up
study begins with a random selection of 40 middle-aged
adults.
a) Find the probability that exactly one-fourth of the
sample suffers from hypertension.
b) Find the mean number of hypertension cases found in
such groups of 40.
c) Find the standard deviation for the numbers of
hypertension cases in groups of 40.
d) What is the probability that less than 4 cases of
hypertension will be found in a group of 40.
e) What is the probability that at least 3 cases of
hypertension will be found in a group of 40.
Answers
1.
Percent
2.
a. 0.0264
b. 0.201
c. 0.9991
d. 20
e. 2
a.
50%
40%
30%
20%
10%
0%
42%
23%
21%
10%
2
3
4
5
3%
1%
6
7 or
more
US Family Size
b. 0.42
c. 0.35
d. 3.12
e. 1.202
3. a. p(x) = .71(.29 )x-1
b. p(x = 1) = .71(.29)1-1 = 0.71
p(x = 2) = .71(.29)2-1 = 0.2059
p(x  3) = 1- p(x = 1) - p(x = 2) = 0.0841
4. a. p(n = 3) = .036(:964)3-1 = 0.0335
p(n = 5) = .036(.964)5-1 = 0.0311
p(n = 12) = .036(:964)12-1 = 0.0241
p(n < 8) = 0.2264
b. 0.8636
5. a. .4043
b. .941
6. a.  = 0.4;
 = 1.02;
2 = 1.04
b.  = 2.6;
 = 0.917;
2 = 0.84
c.  = 3.8;
 = 2.04;
2 = 4.16
d.  = 3;
 = 1.37;
2 = 1.88
e.  = -2.2;
 = 1.37;
2 = 1.88
f.  = 0.8;
 = 1.44;
2 = 2.08
g.  = 0.8;
 = 2.04;
2 = 4.16
h.  = 0;
 = 1.44;
2 = 2.08
i.  = 12.2;
 = 5.016;
2 = 25.16
7. a. 0.9245
b. 0.5951
8. a. 0.8; 0.16; 0.032
b. 0.00032
9. a. 0.04; 0.0384; 0.036864
b. 0.815
10. a. p(n) = 0.3(0.7)n-1
b. 0.147; 0.21
c. 0.2401
11. a. < 0.0001; > 0.9999
b.  = 60;  = 6.481
c. 0.147; 0.0404
12. a. 0.24576
b. 0.262
c. 0.9011
13. a. 0.302
b. 0.322
c. 0.966
14. a. 0.264
b. 0.633
c. 0.367
d. 0.736
15.
x p(x)
0 0.03125
1 0.15625
2 0.3125
3 0.3125
4 0.15625
5 0.03125
16. a. 0.0975
b. 0.0429
c. 0.8145
17.  = 200,  = 13.416
18. a. 0.722
b. 0.484
c.7.5, 2.2913
19. a. Binomial
b. 20
c. 16; 4
20. (a) Discrete
(b) Continuous
(c) Continuous
(d) Discrete
(e) Continuous
(f) Discrete
21. (a) Continuous
(b) Discrete
(c) Discrete
(d) Continuous
(e) Continuous
(f) Discrete
22. (a)
x
1st
2nd
3rd
OT
P(x) 0.309 0.354 0.328 0.009
(b) Goals by Game Period for NHL 89-90
0.4
0.3
0.2
0.1
0
1
23. 0.0127
2
3
4 (OT)
24. a. and b.
Percent
c. 0.673
d. 0.351
e. 41.288
f. 2.326
25. a.
50%
40%
30%
20%
10%
0%
44%
36%
15%
0
1
Number of Fish
b. 0.56
c. 0.20
d. 0.82
e. 0.899
26. a.
b. 0.34
c. 0.58
d. 4.618
e. 2.166
2
4%
1%
3
4 or
more
27. a. 0.746; 1 – 0.254
b. 0.346
c. 0.013
d.  = 1.199
e.  = 0.955
28.
x
P(x)
-80000 0.005
-40000 0.01
-20000 0.05
0
0.935
 = 1800 + 250 = 2050
29. a. 30/3750; 3720/3750
b. 3000*30/3750 = 24; less; 45-24=21
30. a. geometric
b. 0.09
c. 0.729
31. a. 0.0006
b. 3.12
c. 1.696
d. 0.6194
e. 0.6129