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Exercises
Author(s)
David M. Lane
Prerequisites
All material presented in the Probability Chapter
Selected answers
You may want to use the Binomial
Calculator for some of these exercises.
- (a) What is the probability of rolling a pair of dice and
obtaining a total score of 9 or more? (b) What is the probability
of rolling a pair of dice and obtaining a total score of 7?
(relevant section)
- A box contains four black pieces of cloth, two striped
pieces, and six dotted pieces. A piece is selected randomly
and then placed back in the box. A second piece is selected
randomly. What is the probability that:
- both pieces are dotted?
- the first piece is black and
the second piece is dotted?
- one piece is black and one
piece is striped?
(relevant section)
- A card is drawn at random from a deck. (a) What is the
probability that it is an ace or a king? (b) What is the probability
that it is either a red card or a black card? (relevant section)
- The probability that you will win a game is 0.45. (a) If
you play the game 80 times, what is the most likely number of wins?
(b) What are the mean and standard deviation of a binomial
distribution with π = 0.45 and N = 80? (relevant
section)
- A fair coin is flipped 9 times. What is the probability
of getting exactly 6 heads?
(relevant section)
- When Susan and Jessica play a card game, Susan wins 60%
of the time. If they play 9 games, what is the probability
that Jessica will have won more games than Susan? (relevant section)
- You flip a coin three times. (a) What is the probability
of getting heads on only one of your flips? (b) What is the
probability of getting heads on at least one flip? (relevant section & relevant section)
- A test correctly identifies a disease in 95% of people
who have it. It correctly identifies no disease in 94% of people
who do not have it. In the population, 3% of the people have
the disease. What is the probability that you have the disease
if you tested positive? (relevant section)
- A jar contains 10 blue marbles, 5 red marbles, 4 green
marbles, and 1 yellow marble. Two marbles are chosen (without
replacement). (a) What is the probability that one will be
green and the other red? (b) What is the probability that one
will be blue and the other yellow? (relevant section)
- You roll a fair die five times, and you get a 6 each
time. What is the probability that you get a 6 on the next
roll? (relevant section)
- You win a game if you roll a die and get a 2 or a 5.
You play this game 60 times.
- What is the probability that
you win between 5 and 10 times (inclusive)?
- What is the probability that you will win the game
at least 15 times?
- What is the probability that you
will win the game at least 40 times?
- What is the most likely number of wins.
- What is the probability of obtaining the number of wins
in d?
(relevant
section)
- In a baseball game, Tommy gets a hit 30% of the time
when facing this pitcher. Joey gets a hit 25% of the time.
They are both coming up to bat this inning.
- What is the
probability that Joey or Tommy (but not both) will get a hit?
- What
is the probability that neither player gets a hit?
- What is the probability that they both get a hit? (relevant
section)
- An unfair coin has a probability of coming up heads of
0.65. The coin is flipped 50 times. What is the probability
it will come up heads 25 or fewer times? (Give answer to at
least 3 decimal places). (relevant section)
- You draw two cards from
a deck, what is the probability that
- both of them are
face cards (king, queen, or jack)?
- What is the probability
that you draw two cards from a deck and both of them
are hearts? (relevant
section)
- True/False: You are more likely to get a pattern of HTHHHTHTTH
than HHHHHHHHTT when you flip a coin 10 times. (relevant
section)
- True/False: Suppose that at your regular physical exam
you test positive for a relatively rare disease. You will need
to start taking medicine if you have the disease, so you ask
your doctor about the accuracy of the test. It turns out that
the test is 98% accurate. The probability that you have Disease
X is therefore 0.98 and the probability that you do not have
it is .02. (relevant section)
Questions from Case Studies:
The following questions are from the Diet and Health (DH)
case study.
- (DH#1)
- What percentage of people on the AHA diet
had some sort of illness or death?
- What is the probability
that if you randomly selected a person on the AHA diet,
he or she would have some sort of illness or death? (relevant
section)
- If 3 people on the AHA diet are chosen at random,
what is the probability that they will all be healthy?
(relevant section)
- (DH#2)
- What percentage of people on the Mediterranean
diet had some sort of illness or death?
- What is the probability that if you randomly selected
a person on the Mediterranean diet, he or she would have
some sort of illness or death? (relevant
section)
- What is the probability that if you randomly
selected a person on the Mediterranean diet, he or she
would have cancer? (relevant section)
- If
you randomly select five people from the Mediterranean
diet, what is the probability that they would all be
healthy? (relevant section)
The following questions are from (reproduced with permission)
Visit the site
- Five faces of a fair die are painted black, and one face
is painted white. The die is rolled six times. Which of the
following results is more likely?
a. Black side up on five of the rolls; white side up on the other roll
b. Black
side up on all six rolls
c. a and b are equally likely
- One of the items on the student survey for an introductory
statistics course was "Rate your intelligence on a scale
of 1 to 10." The distribution of this variable for the
100 women in the class is presented below. What is the probability
of randomly selecting a woman from the class who has an intelligence
rating that is LESS than seven (7)?
a. (12 + 24)/100 = .36
b. (12 + 24 + 38)/100 = .74
c. 38/100 = .38
d. (23 + 2 + 1)/100 = .26
e. None of the above.
- You roll 2 fair six-sided dice. Which of the following
outcomes is most likely to occur on the next roll? A. Getting
double 3. B. Getting a 3 and a 4. C. They are equally likely.
Explain your choice.
- If Tahnee flips a coin 10 times, and records the results
(Heads or Tails), which outcome below is more likely to occur,
A or B? Explain your choice.
- A bowl has 100 wrapped hard candies in it. 20 are yellow,
50 are red, and 30 are blue. They are well mixed up in the
bowl. Jenny pulls out a handful of 10 candies, counts the number
of reds, and tells her teacher. The teacher writes the number
of red candies on a list. Then, Jenny puts the candies back
into the bowl, and mixes them all up again. Four of Jenny's
classmates, Jack, Julie, Jason, and Jerry do the same thing.
They each pick ten candies, count the reds, and the teacher
writes down the number of reds. Then they put the candies back
and mix them up again each time. The teacher's list
for the number of reds is most likely to be (please select
one):
a. 8,9,7,10,9
b. 3,7,5,8,5
c. 5,5,5,5,5
d. 2,4,3,4,3
e. 3,0,9,2,8
- An insurance company writes policies for a large number
of newly-licensed drivers each year. Suppose 40% of these are
low-risk drivers, 40% are moderate risk, and 20% are high risk.
The company has no way to know which group any individual driver
falls in when it writes the policies. None of the low-risk
drivers will have an at-fault accident in the next year, but
10% of the moderate-risk and 20% of the high-risk drivers will
have such an accident. If a driver has an at-fault accident
in the next year, what is the probability that he or she is
high-risk?
- You are to participate in an exam for which you
had no chance to study, and for that reason cannot do anything but guess
for each question (all questions being of the multiple choice
type, so the chance of guessing the correct answer for each
question is 1/d, d being the number of options
per question; so in case of a 4-choice question, your
chance is 0.25). Your instructor offers you the opportunity
to choose amongst the following exam formats: I. 6 questions
of the 4-choice type; you pass when 5 or more answers are correct;
II. 5 questions of the 5-choice type; you pass when 4 or more
answers are correct; III. 4 questions of the 10-choice type;
you pass when 3 or more answers are correct. Rank the three
exam formats according to their attractiveness. It should be
clear that the format with the highest probability to pass
is the most attractive format. Which would you choose and why?
- Consider the question of whether the home team wins
more than half of its games in the National Basketball Association.
Suppose that you study a simple random sample of 80 professional
basketball games and find that 52 of them are won by the home
team.
- Assuming that there is no home court advantage and
that the home team therefore wins 50% of its games in the
long run, determine the probability that the home team
would win 65% or more of its games in a simple random sample
of 80 games.
- Does the sample information (that 52 of a random
sample of 80 games are won by the home team) provide strong
evidence that the home team wins more than half of its
games in the long run? Explain.
- A refrigerator contains 6 apples, 5 oranges, 10 bananas,
3 pears, 7 peaches, 11 plums, and 2 mangos.
- Imagine you stick your hand in this refrigerator and
pull out a piece of fruit at random. What is the probability
that you will pull out a pear?
- Imagine now that you put your hand in the refrigerator
and pull out a piece of fruit. You decide you do not want
to eat that fruit so you put it back into the refrigerator
and pull out another piece of fruit. What is the probability
that the first piece of fruit you pull out is a banana
and the second piece you pull out is an apple?
- What is the probability that you stick your hand
in the refrigerator one time and pull out a mango or an
orange?
Answers:
1) (a) 5/18 = .278
2) (b) 1/6 = .167
3) (a) 2/13 = .154
4) (b) mean = 36; SD = 4.24
5) .164
7) (b) 7/8 = .875
9) (a) 2/19 = .105
11) (b) 0.937
12) (c) .075
14) (b) 1/17 = .0588
17) (c) .493
18) (b) .10
Please answer the questions:
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