MT004.04
Review problems for Midterm 1
1. Draw a two-circle Venn diagram, and shade the portion corresponding to the set (S [ T 0 )0 .
2. There are 16 contestants in a tennis tournament. How many di↵erent possibilities are there for the two
people who will play in the final round?
3. In how many ways can a coach and five basketball players line up in a row for a picture if the coach insists
on standing at one of the ends of the row?
4. Draw a three-circle Venn diagram. and shade the portion corresponding to the set R0 \ (S [ T ).
5. An urn contains 14 numbered balls, of which 8 are red and 6 are green. How many di↵erent possibilities
are there for selecting a sample of 5 balls in which 3 are red and 2 are green?
6. There are 12 contestants in a contest. Two will receive trips around the world, four will receive cars, and
six will receive TV sets. In how many di↵erent ways can the prizes be awarded?
7. The 100 members of the Earth Club were asked what they felt the club’s priorities should be in the
coming year: clean water, clean air, or recycling. The responses were 45 for clean water, 30 for clean air,
42 for cycling, 13 for both clean air and clean water, 20 for clean air and recycling, 16 for clean water and
recycling, and 9 for all three.
(a) How many members thought that the priority should be clean air only?
(b) How many members thought that the priority should be clean water or clean air, but not both?
(c) How many members thought that the priority should be clean water or recycling, but not clean air?
(d) How many members thought that the priority should be Clean air and recycling but not clean water?
(e) How many members thought that the priority should be exactly one of the three issues?
(f) How many members thought that recycling should not be a priority?
(g) How many members thought that the priority should be recycling but not clean air?
(h) How many members thought that the priority should be something other than one of these three
issues?
8. How many di↵erent nin-letter words (sequence of letters) can be made by using four Ss and five Ts?
9. Twenty people take an exam. How many i↵erent possibilities are there for the set of people who pass the
exam?
10. How many di↵erent meals can be chosen if there are 6 appetizers, 10 main dishes, and 8 desserts, assuming
that a meal consists of one item from each category?
11. How many seven-digit numbers are even and have a 3 in the hundreds place?
12. How many strings of length 8 can be formed from the letters A, B, C, D, and E? How many of the strings
have at least one E?
13. In how many ways can five people be assigned to seats in a 12-seat room?
14. How many three-digit numbers are there in which exactly two digits are alike?
15. In how many ways can a selection of at least one tie be made from a set of eight ties?
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MT004.04
Review problems for Midterm 2
1. A box contains a penny, a nickel, a dime, a quarter, and a half dollar. You select two coins at random.
(a) Write the elements of the sample space for this experiment.
(b) List the elements of the event E in which the total value of the coins you have selected is an even
number of cents.
2. Some of the candidates for president of the computer club at Riverdale High are seniors, and the rest are
juniors. Let J be the event in which a junior is elected, and F be the event in which a female is elected.
Describe the following events:
(a) J \ F 0
(b) (J \ F )0
(c) J [ F 0
3. Suppose that E and F are mutually exclusive events with Pr(E) = .5 and Pr(F ) = .3. Find Pr(E [ F ).
4. Of the 120 students in a class, 30 speaks Chinese, 50 speak Spanish, 75 speak French, and 12 speak Spanish
and Chinese, 30 speak Spanish and French, and 15 speak Chinese and French. Seven students speak all
three languages. A student is chosen at random. What is the probability that the chosen student speaks
none of these languages?
5. A committee consists of five men and five women. If three people are selected at random from the
committee, what is the probability that they all will be men?
6. Five of the apples in a barrel of 100 apples are rotten. If four apples are selected from the barrel, what is
the probability that at least two of the apples are rotten?
7. Of the nine city council members, four favor the school vouchers and five are opposed. If a subcommittee
of three council members is selected at random, what is the probability that exactly two of them favor
school vouchers?
8. A coin is to be tossed five times. What is the probability of obtaining at least one head?
9. Two players each toss a coin three times. What is the probability that they get the same number of tails?
10. In an Olympic swimming event, two of the seven contestants are American. The contestants are randomly
assigned to lanes 1 through 7. What is the probability that the Americans are assigned to the first two
lanes?
11. A collection of code words consists of all strings of seven characters, where each of the first three characters
can be any letter or digit and each of the last four characters must be a digit. For example, AA32765 is
allowed, but 7A3B765 is not.
(a) What is the probability that a code word chosen at random begins with ABC?
(b) What is the probability that a code word chosen at random ends with 6578?
(c) What is the probability that a code word chosen at random consists of three letters followed by four
even digits?
12. A card is drawn at random from a standard deck of 52 cards. Then the card is replaced, and the deck is
throughly shu✏ed. This process is repeated two times.
three
(a) What is the probability that all three cards are aces?
(b) What is the probability that at least one of the cards is an ace?
13. What is the probability of having each of the numbers one through six appear in six consecutive rolls of
a die?
14. What is the probability that out of a group of five people, at least two people have the same birthday?
Assume that there are 365 days in a year.
15. Four people chosen at random. What is the probability that at least two of them were born on the same
day of the week?
16. Let E and F be events with Pr(E \ F ) =
1
10
and Pr(E|F ) = 17 . Find Pr(F ).
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17. When a coin is tossed three times, what is the probability of at least one tail appearing, given that at
least one head appeared?
18. Suppose that a pair of dice is rolled. Given that the two numbers are di↵erent, what is the probability
that one die shows a three?
19. Out of 50 colleges in a certain state, 25 are private, 15 o↵er engineering majors, and 5 are private colleges
o↵ering engineering majors. Find the probability that a college chosen at random from the state
(a) o↵ers an engineering major.
(b) o↵ers an engineering major given that it is public.
(c) is private given that it o↵ers an engineering major.
(d) is public, given that it o↵ers an engineering major.
20. An urn contains 10 red balls and 20 green balls. If four balls are drawn one at a time without replacement,
what is the probability that the sequence of colors will be red, green, green, red?
21. Two archers shoot at a moving target. One can hit the target with probability 1/4 and the other with
probability 1/3. Assuming that their e↵orts are independent events, what is the probability that
(a) both will hit the target?
(b) at least one will hit the target?
22. Each box of a certain brand of candy contains either a toy airplane and a toy boat. If one-third of the
boxes contain an airplane and two-thirds contain a boat, what is the probability that a person who buys
two boxes of candy will receive both an airplane and a boat?
23. An urn contains three balls numbered 1, 2, and 3. Balls are drawn ane at a time without replacement
until the sum of the numbers drawn is four or more. Find the probability of stopping after exactly two
balls are drawn?
24. According to a geneticist at Stanford University, the chances of having a left-handed child are 4 in 10 if
both parents are left-handed, 2 in 10 if one parent is left-handed, and only 1 in 10 if neither parent is
left-handed. Suppose that a left-handed child is chosen at random from a population in which %25 of the
adults are left-handed. What is the probability that the child’s parents are both left-handed?
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MT004.04
Review problems for Midterm 3
1. Find the five-number summary and the interquartile range for the following set of numbers, and then
draw the box plot:
1, 2, 3, 4, 5, 9, 14, 23
2. The manager of a supermarket counts the number of customers waiting in the express check-out line at
random times throughout the week. Her observations are found in the following frequency table:
Number waiting in line
0
1
2
3
4
5
6
Frequency
2
5
9
13
11
7
3
Construct the corresponding relative frequency table, and use it to estimate the probability that at most
three customers are waiting in line.
3. A fair coin is tossed twice. Let X be the number of heads.
(a) Determine the probability distribution of X.
(b) Determine the probability distribution of 2X + 5.
4. An experiment consists of three binomial trials, each having probability 1/3 of success.
(a) Determine the probability distribution table for the number of successes.
(b) Use the table to compute the mean and the variance of the probability distribution.
5. An archer has probability .3 of hitting a certain target. What is the probability of hitting the target
exactly two times in four attempts?
6. An urn contains four red balls and four white balls. An experiment consists of selecting at random
a sampple of four balls and recording the number of red balls in the sample. Set up the probability
distribution, and compute its mean and variance.
7. The probability distribution of a random variable X is given below. Determine the mean, variance, and
standard deviation of X.
k Pr(X=k)
-2
.3
0
.1
1
.4
3
.2
8. Lucy and Ethel play a game of chance in which a pair of dice is rolled once. If the result is 7 or 11, then
Lucy pays Ethel $10. Otherwise, Ethel pays Lucy $3. In the long run, which player comes out ahead, and
by how much?
9. Suppose that a probability distribution has mean 10 and standard deviation 1/3. Use Chebychev inequality
to estimate the probability that an outcome will lie between 9 and 11.
10. Suppose X is a random variable having a normal distribution with µ = 5 and
Find Pr(6.5  X  11).
= 3.
11. The height of adult males in United States is normally distributed with µ = 5.75 feet and
What percent of the adult male population has height of 6 feet or greater?
= .2 feet.
12. In a certain city, two-fifths of the registered voters are women. Out of a group of 54 voters allegedly
selected at random for jury duty, 13 are women. A local civil liberties group has charged that the
selection procedure discriminated against women. Use the normal curve to estimate the probability of 13
or fewer women being selected in a truely random selection process.
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13. In a complicated production process, 1/4 of the items produced have to be readjusted. Use the normal
curve to estimate the probability that out of a batch of 75 items, between 8 and 22 (inclusive) of the items
require readjustment.
14. Costruct truth tables for the given statement forms
(a) ⇠ [(p ^ r) _ q]
(b) ⇠ ((p _ q) ^ (^q))
15. Compare the truth tables of ⇠ p _ ⇠ q and ⇠ (p ^ q).
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