MATH 1530 Unit 2 Packet S20
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MATH 1530
Unit 2 Formula Sheet
Chapter 3
Classical (or Theoretical) Probability:
Empirical (or Statistical) Probability:
Probability of a Complement:
Probability of occurrence of both events A and
B:
if are independent
Probability of occurrence of either A or
B:
if A and B are mutually exclusive
Chapter 4
Mean of a Discrete Random Variable:
Variance and Standard Deviation of a Discrete Random Variable:
Variance: Standard Deviation: Expected Value:
Binomial Probability of x successes in n trials:
Population Parameters of a Binomial Distribution:
Mean: Variance: Standard Deviation: 1
MATH 1530
Unit 2 Probability
3.1 Basic Concepts of Probability and Counting
Basic Probability
Calculating the Probability of an Event
1. In a bag, you have
You randomly select one chip from the bag. Find:
Event
Theoretical Probability
Experimental Probability
P(B)
P(R)
P(G)
P(4)
P(3)
P(
) = P(not blue)
P(
)
P(
)
P(number less than 5)
P(5)
4 blue chips numbered 1, 2, 3, & 4
3 red chips numbered 1, 2, & 3
2 green chips numbered 1 & 2
2
Complements Rule
What do you notice about P(
) and P(
)? What do you notice about P(
) and P(
)? Probability Rules for Complements
1. 2. 3. Some Special Probabilities
Impossible Event: Certain Event: Unusual Event: 2. You roll a fair 6-sided die
a. List the Sample Space:
b. What is the probability of:
i. rolling an odd number greater than 1?
ii. not rolling an odd number greater than one?
iii. rolling a value greater than 6?
iv. rolling a value between 1 and 6, inclusive?
3
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Creating the List of Outcomes for a Large Sample Space
3. You are ordering salad at Panera. You will select between 4 salads as your entrée. The 4
salads are: Chicken Cobb with Avocado, Thai Chicken, Strawberry Poppyseed with Chicken,
and Fuji Apple Chicken. You have three choices for side items: chips, apple or baguette. List
the different outcomes for your sample space if your meal will consist of one salad and one
side item.
a. How many possible outcomes do we have in our sample space? b. You decide to add a drink. Your choices are: Coke, Diet Coke, Sprite, Sweet Tea and
UnSweet Tea. Now how many outcomes are in your sample space?
Cobb Salad and Chips
Cobb Salad and Apple
Cobb Salad and Baguette
Thai Salad and Chips
Thai Salad and Apple
Thai Salad and Baguette
Strawberry Salad and Chips
Strawberry Salad and Apple
Strawberry Salad and Baguette
Fuji Salad and Chips
Fuji Salad and Apple
Fuji Salad and Baguette
4
Fundamental Counting Principle:
For a sequence of two events in which the first event can occur in m ways and the second
event can occur in n ways, the events together can occur in m*n ways. This rule can be
extended to any number of events occurring in sequence.
4. You are choosing an outfit to wear to school. You have 10 shirts, 5 pairs of pants, and 3 pairs
of shoes. How many outfits can you make?
5. An access code consists of six characters. For each character, any letter or number can be
used, with the exceptions that the first character cannot be 0, and the last two characters
must be odd numbers.
6. What is the probability of randomly selecting the correct access code on the first try? 5
MATH 1530
Unit 2 Probability
3.2 Conditional Probability and the Multiplication Rule
Open the Global Health data set, and use it to complete the table below with the correct number
of countries for each cell:
Average Income ≤ $28000
Average Income > $28000
Total Number
Life Expectancy ≤ 79
Life Expectancy >79
Total
The table you created above is called a contingency table. It has that name because it shows how
the individuals are distributed along each variable, contingent on the value of the other variable.
We will use this table to calculate some probabilities.
Does there appear to be a relationship between the wealth and life expectancy of a country? Does this seem to be a representative sample from which to make conclusions? The Multiplication Rule
is called a conditional probability. It represents the probability of event B occurring
AFTER it is assumed event A has already occurred.
The Multiplication Rule with Independent Events
The occurrence of Event A does not affect the probability that Event B will occur and the
occurrence of Event B does not affect the probability that Event A will occur. You may see the
words “with replacement” when finding the probability of selecting multiple items.
When A and B are independent events,
. This means you can simplify the
multiplication rule to:
The Multiplication Rule with Dependent Events
When A and B are dependent events, the occurrence of Event A affects the probability that Event
B will occur and the occurrence of Event B affects the probability that Event A will occur. You may
see the words “without replacement” when finding the probability of selecting multiple items.
6
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When A and B are dependent events,
must be used as the probability Event B will
occur. We have assumed Event A has occurred and Event A affects the probability that Event B
will occur. You must use the original multiplication rule:
The Multiplication Rule with Contingency Tables (Dependent and Independent Events)
Average Income ≤ $28000
Average Income > $28000
Total Number
Life Expectancy ≤ 79
Life Expectancy >79
Total
1. If you randomly select one country, what is the probability of ...?
a. P(Life Expectancy >79) = b. P(Life Expectancy ≤ 79 and Average Income > $28000) = 2. Independent Compound Events:
If you randomly select two countries, with replacement
,
what is the probability of ...?
a. P(Both Average Income > $28000) = b. P(Average Income > $28000, and then Life Expectancy >79) = 3. Dependent Compound Events:
If you randomly selected two countries, without
replacement
, what is the probability of ...?
P(Both Average Income > $28000) = 4. Conditional Probability
What is the percentage breakdown by Life Expectancy if we separate the wealthier and
poorer countries?
a. P(Life Expectancy ≤ 79, given that Average Income > $28000) = b. Another notation for this : P(Life Expectancy ≤ 79 ǀ Average Income > $28000) = c. P(Life Expectancy ≤ 79, given that Average Income ≤ $28000) = d. Another notation for this : P(Life Expectancy ≤ 79 ǀ Average Income ≤ $28000) =
7
e. When we limit ourselves to a specific column or row, so that we can explore the
distribution of one variable based on whether they meet the criteria of another
variable, we have created a conditional distribution. Why do you think it got the
name “conditional”?
5. Consider an illness that has a prevalence of about 1 in 1000, and the test for this illness has
a 95% accuracy (meaning that patients who have the illness test positive 95% of the time).
The results for a large number of patients tested would look something like this:
Tests negative
Tests positive
Total Number
Has illness
1
19
20
Does not have illness
18981
999
19980
Total
18982
1018
20000
a. If someone tests positive, what is the probability that they actually have the illness?
b. If someone tests negative, what is the probability that they actually have the illness?
c. If someone tests positive, what is the probability they DO NOT have the illness?
d. Why would it be important for doctors to understand these probabilities?
8
6. The following contingency table represents a bowl full of M&Ms and Skittles. Find the
probabilities:
Red
Candy
Yellow
Candy
Green
Candy
Blue
Candy
Orange
Candy
Brown or
Purple
Total
Number
M&Ms
109
102
142
183
187
105
828
Skittles
202
188
145
0
154
156
845
Total
Number
311
290
287
183
341
261
1673
a. One randomly selected a piece of candy:
i. ii. iii. iv. b. Two randomly selected pieces of candy:
i. with replacement ii. without replacement iii. with replacement iv. without replacement
9
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7. Republicans
Democrats
Independents
Total Number
Males
46
39
1
86
Females
5
9
0
14
Total Numer
51
48
1
100
a. Randomly select 1 person
i. P (Female | Democrat) ii. P (Republican | Female) b. Randomly select 2 people:
i. P (Both Female) with replacement ii. P (Both Female) without replacement iii. P (Both Democrat) with replacement iv. P (Both Democrat) without replacement v. P(Democrat then Republican) without replacement vi. P(Democrat then Republican) with replacement 8. Example with Scientific Notation:
a. You have 20,000 CD’s in a warehouse. 2% are defective. You choose 3 CD’s from the
20,000.
b. P (3 CD’s are defective) with replacement =
c. P (3 CD’s are defective) without replacement =
Some calculations are cumbersome, but they can be made manageable by using the common
practice of treating events as independent when small samples are drawn from large populations.
In such cases, it is rare to select the same item twice.
10
The 5% Guideline for Cumbersome Calculations:
If a sample size is no more than 5% of the
size of the population, treat the selections as being independent (even if the selections are made
without replacement, so they are technically dependent).
So, in the example above, we could treat the events as independent, even though the CDs are
being selected “without replacement”.
Practice with Scientific Notation
Write the following in Exponential Notation and Scientific Notation:
1. 3.56 E -4 2. 7.83 E 5 11
MATH 1530
Unit 2 Probability
3.3 The Addition Rule
Compound Events:
Combining two or more simple events.
Mutually Exclusive (or Disjoint) events:
Compound events that CANNOT OCCUR at the same
time
The Addition Rule: Probability of Event A OR
Event B occurring.
1. In our bag of poker chips, what was P(B)?
P(1)?
What is P(
Blue or
1)?
To calculate P(
Blue or
1), we will need P(
Blue and
1): 12
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2. Let's roll 2 six-sided dice. 1 1
1 2
1 3
1 4
1 5
1 6
2 1
2 2
2 3
2 4
2 5
2 6
3 1
3 2
3 3
3 4
3 5
3 6
4 1
4 2
4 3
4 4
4 5
4 6
5 1
5 2
5 3
5 4
5 5
5 6
6 1
6 2
6 3
6 4
6 5
6 6
a. P (sum of 3) = b. P (sum of 7) = c. P (sum of 1) = d. P (sum < 13 ) = e. P (first die is 3) = f. P(second die is 5) = g. P (first is a 3 or second is a 5) = 3. With a standard deck of cards and a fair die, find the following probabilities:
a. P(draw a heart and roll a 3)
b. P(draw a face card and roll an odd number)
c. P(draw a jack, given that you draw a face card)
d. P(draw a heart or roll a three)
e. P(draw a red or draw a ten)
f. P(draw a king or draw a diamond)
13
4. In a bag, you have
You randomly select one chip from the bag. Find:
a. P (blue or red)
b. P (blue or 3) c. P (red or 3) d. P (green or 3) 4 blue chips numbered 1, 2, 3,and 4
3 red chips numbered 1, 2, and 3
2 green chips numbered 1 and 2.
14
Using the Addition Rule with a Contingency Table
5. The following table gives the number of people with health insurance coverage in the U.S. in
2018. Note that the numbers are in millions of people.
Source of data
(In
millions)
Private
insurance
Private
insurance
(Marketplace)
Public
Insurance
Uninsured
Total
Number
Adults
age 18-
64
128.2
8.4
38.5
26.4
Children
age 0-
17
38.5
1.7
30.7
3.8
Total
Number
Find the probabilities if you randomly select one person:
a. P(Adult) = b. P(Uninsured) = c. P(Adult and Uninsured) = d. P(Adult or Uninsured) =
Group Discussion:
e. P(Child) = f. P(public insurance) = g. P(private insurance) = h. P(Child or private insurance) = i. P(Child or no private insurance)
15
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MATH 1530
Unit 2 Probability
4.1 Probability Distributions
Human blood comes in different types. Each person has a specific ABO type (A, B, AB, or O) and
Rh factor (positive or negative). Hence, if you are O+, your ABO type is O and your Rh factor is
positive. A probability model for blood types in the United States is given:
1. Blood Type
A+
A-
B+
B-
AB+
AB-
O+
O-
Probability
0.357
0.063
0.085
0.015
0.034
0.006
0.374
0.066
a. What is the probability that a randomly chosen person has blood type A?
b. What is the probability that two randomly chosen U.S. residents both have type A
blood?
c. What is the probability that at least one of the two people does not have type A
blood?
A probability distribution is what will probably happen, not necessarily what actually
happened. Probability distributions are theoretical, not experimental. These distributions are
usually given in the form of a graph, table or formula.
A random variable, x, represents a value associated with each outcome of the distribution.
Qualifiers for probability distributions:
All probabilities are between impossible and certain, inclusive.
2. Do the following distributions meet both requirements? Why/Why not?
Distribution A:
x values
Probability P(x)
2
0.4
3
1.2
5
0.1
Distribution B:
x values
Probability P(x)
2
0.4
3
0.4
5
0.2
Distribution C:
x values
Probability P(x)
2
0.1
3
0.6
5
0.1
16
Two Types of Random Distributions: Discrete and Continuous
Discrete Distributions:
A random variable is discrete when it has a finite or countable number of possible outcomes
that can be listed. A discrete distribution lists each possible outcome for x together with the
probability of x occurring. The probability distribution is a relative frequency distribution.
Continuous Distributions:
A random variable is continuous when it has an uncountable number of possible outcomes,
represented by an interval on a number line.
3. Determine whether each random variable, x, is discrete or continuous. Explain your
reasoning.
a. Let x represent the number of cars in the PSCC Hardin Valley parking lots.
b. Let x represent the height of a random student on PSCC campus.
Creating a Discrete Probability Distribution
4. You take a 2 question quiz. The first question, Q1, is T/F and the second question, Q2, is
multiple choice with a, b, c, or d as your choices. You guess on both. After you submit your
quiz, what are the following probabilities?
a. P (Q1 is correct) b. P (Q2 is correct) c. P (Both are Correct) = P (C and C ) d. P (C and W) e. P (W and C) f. P (W and W) g. Use the information in a-f to create a probability distribution table:
Let x = number correct
x values
Probability P(x)
0
1
2
h. Does our distribution meet both requirements for a probability distribution?
17
5. Use the data in the table below to answer the questions
y
1
2
3
4
5
6
7
8
p(y)
0.15
0.23
0.19
0.23
0.12
0.05
0.02
0.01
a. What is the probability that a randomly selected family will consist of at least two
people?
b. What is the probability that a randomly selected family will have two to four
members?
c. What is the mean family size?
d. What is the standard deviation for family size?
e. Would selecting a family with 8 members be unusual?
18
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6. A cell phone service has the following distribution of cell phones associated with each
friends and family account.
(The symbol 0+ denotes a positive probability value that is very small. We are not willing to
say this is an impossible event. Use P(x=6) = 0 in your calculations.)
x values
Probability P(x)
1
.02
2
.13
3
.32
4
.39
5
.14
6
0+
If one account is randomly selected, find the following probabilities for the number of
phones associated with the account:
a. Probability of exactly 3 phones:
b. Probability of 4 or more phones:
c. Probability of at least 2 phones:
d. Probability of at most 3 phones:
e. Probability of no more than 2 phones:
f. Probability of between 2 and 5 phones:
g. Probability of between 2 and 5 phones, inclusive:
19
Mean (Expected Value) and Standard Deviation of a Discrete Probability Distribution
7. A cell phone service has the following distribution of cell phones associated with each
friends and family account.
x values
Probability P(x)
1
.02
2
.13
3
.32
4
.39
5
.14
6
0+
a. Did we keep both rules for probability distributions?
b. MEAN (EXPECTED VALUE):
c. VARIANCE:
d. STANDARD DEVIATION:
e. UNUSUAL VALUES:
Min usual value: Max usual value: Would selecting an account with 1 phone be unusual? Would selecting an account with 2 phones be unusual? Would selecting an account with 5 phones be unusual? 20
MATH 1530
Unit 2 Probability
4.2 Binomial Distributions
Binomial Probability Distributions and Random Variables
Binomial Probability Distributions result from a procedure that meets all the following
requirements:
The procedure has a FIXED number of trials
The trials are INDEPENDENT (the outcome of any individual trial does not affect the
probabilities in the other trials.)
All trial outcomes can be classified into TWO CATEGORIES (success and failure).
The probability of success REMAINS THE SAME throughout the trials.
Calculating Binomial Probabilities
Flip coin 5 times Probability of 4 tails
p = probability of one success = q = probability of one failure q = 1 - p = n = number of trials = x = number of successes WE ARE LOOKING FOR = 1. Are the following binomial? If not, explain why not.
a. Tossing a coin 20 times to see how many tails occur. b. Asking 200 people if they have watched ABC news in the past week. c. Asking 20 people how old they are. d. Rolling a die to see if you roll a 5. e. Rolling a die until a 5 appears. 21
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Probabilities in a Binomial Distribution
2. Assume that a procedure yields a binomial distribution with a trial repeated n times. Find the
probability of x successes given the probability of success (p) on a given trial.
a. n = 14 trials
x = 4 successes
p = .60 probability of success
b. n = 15 trials
x = 13 successes
p = 1/3 probability of success
c. n = 23 trials
x = 12 successes
p = .4 probability of success
22
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3. You toss a fair coin 3 times.
a. Create a probability distribution function (in table form) for the outcomes.
n = p = q = Note: x is the number of heads out of the three tosses. We will need to find P(x) for all
possible outcomes to create the table.
b. What is the probability of tossing AT MOST 2 heads?
c. What is the probability of tossing AT LEAST 2 heads?
P(x = 0) = P(x = 1) = P(x = 2) = P(x = 3) = x values
Probability P(x)
0
1
2
3
23
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4. A quiz has two multiple choice questions each with four answer choices.
a. Create a probability distribution function table for the number of correct questions
n = 2
p = q = x = number correct
x values
Probability P(x)
0
1
2
b. What is the probability you get NO MORE THAN 1 correct? c. What is the probability you get AT LEAST 1 correct?
24
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5. Checking Your Answers with Technology.
In 2017, 92% of American children were
vaccinated against measles. https://data.worldbank.org/indicator/sh.imm.meas
A health worker wants to gather some data to see if the vaccination rate may have changed
recently. She randomly selects 30 children for her study.
n = p = q = a. Find the probability that all
of the children are vaccinated. b. Find the probability that exactly half
of the children are vaccinated.
c. Find the probability that at least half
of the children are vaccinated. d. Find the probability that at most 25
of the children are vaccinated. e. Find the probability that between 25 and 30 children are vaccinated.
f. If at most 25 of the children are vaccinated, does it appear that the 92% vaccination
rate is wrong?
g. Would 24 children out of the 30 being vaccinated be unusual? Why or why not?
25
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6. Using Technology:
In a region, 80% of the population has red hair. If 30 people are
randomly selected, find the following probabilities and identify the unusual events. Use a
significance level of 0.05.
By hand gets complicated: Words
Probability Statement
Calculated Probability
Unusual (Yes or No)
Probability of at least 27 out of 30
Probability of less than 25 out of 30
Probability of no more than 18 out of 30
Probability of more than 21 out of 30
Probability of at least 26 out of 30
Probability of 16 out of 30
Probability of less than 4 out of 30
7. A pharmaceutical company receives large shipments of aspirin tablets. The acceptable
sampling plan is randomly select and test 21 tablets, then accept the whole batch if there is
no more than 1 tablet in the sample that does not meet the required specifications. If a
particular shipment of thousands of aspirin actually has a 3% rate of defects, what is the
probability this whole shipment will be accepted? What is the probability it will be rejected?
26
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Mean, Variance and Standard Deviation of a Binomial Random Variable
Binomial Distribution Center and Spread Formulas:
Mean: Variance: Standard Deviation: Where
n = number of fixed trials
p = probability of success in one of the n trials (
can be given as a percentage OR as a
probability)
q = probability of failure in one of the n trials, q = 1 - p
8. McDonald’s has a 95% recognition rate. A special focus group consists of 12 randomly
selected adults.
a. For such a group, find the mean, variance, and standard deviation.
Mean: Variance: Standard Deviation: b. Use the range rule of thumb to find the minimum and maximum usual number of
people who would recognize McDonald’s.
Minimum: Maximum: 9. Suppose that Bayanisthol, a new drug, is effective for 65% of the participants in clinical trials.
If a group of fifteen patients take this new drug,
a. What is the expected number of patients for whom the drug will be effective? b. What is the probability that the drug will be effective for less than half of them?
27
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MATH 1530
Unit 2 Probability
Review
1. On an ACT or SAT test, a typical multiple-choice question has 5 possible answers. If you
make a random guess on one such question, what is the probability that your response is
wrong?
2. The General Motors Corporation wants to conduct a test of a new model of Corvette. A pool
of 200 drivers has been recruited, 80 of whom are men. When the first person is selected
from this pool, what is the probability of not getting a male driver?
3. Two psychologists surveyed 478 elementary children in Michigan. They asked students
whether their primary goal was to get good grades, to be popular, or to be good at sports.
Below is a contingency table giving counts of the students by their goals and demographic
setting of their school: rural, suburban, or urban.. Complete the table and answer the
questions.
Grades Most Important
Popularity Most Important
Sports Most Important
Total Number
Rural
57
50
42
Suburban
87
42
22
Urban
103
49
26
Total Number
For one randomly selected student, find:
a. P(Rural) = b. P(Sports) = c. P(Suburban and Popularity) = d. P(Suburban or Popularity) = e. P(Popularity | Rural) = f. P(Rural | Grades) = 28
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4. According to cancer.org, there is about a 25% success rate for those who try to stop
smoking through medication alone. Find the probability that for 8 randomly selected
smokers who use medication, they all successfully quit smoking.
5. If Knoxville has an annual robbery rate of 0.23%, find the probability that among 3 randomly
selected residents, all have been robbed during a given year. (The population of Knoxville is
about 183,000.)
Data from http://www.neighborhoodscout.com/tn/knoxville/crime/
6. Find the probability of drawing a hand of cards that is 2 Kings 3 Aces (not replacing the
cards) from a standard deck.
7. You roll two dice three times. Find the probability of rolling a seven all three times.
8. Below is a table showing the number of people with various levels of education in 5
countries.
Post-
graduate
Some
College
Some High
School
Primary or
Less
No
Answer
Total
Number
China
7
315
671
506
3
1502
France
69
388
766
309
7
1539
India
161
514
622
227
11
1535
U.K.
58
207
1240
32
20
1557
USA
84
486
896
87
4
1557
Total
Number
379
1910
4195
1161
45
7690
If we select someone randomly from the survey, what is the probability the person…
a. is from the US? b. completed his or her education before college? c. is from France or did some post-graduate study?
29
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9. If I roll a regular fair die, what is the probability of rolling an even number or a number
greater than 2?
10. a. What is the probability of randomly selecting a 6 or a club out of a deck of cards?
b. What is the probability of randomly selecting a queen or a 4 out of a deck of cards?
c. What is the probability of randomly selecting a face card or a diamond out of a deck
of cards?
11. Complete the statement: 12. What is the probability of randomly selecting a red pen OR a black pen from a box
containing 6 red pens, 3 black pens, and 2 blue pens?
13. Eye Color: Groups of 5 babies are randomly selected. In each group, the random variable x is
the number of babies with green eyes.
x values
Probability P(x)
0
0.528
1
0.360
2
0.098
3
0.013
4
0.001
5
0+
a. Does it meet the requirement for a probability distribution? Explain
b. Expected Value: E(x) = show work for credit
Mean c. Variance:
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Standadard Deviation d. Minimum Usual Value: Maximum Usual Value: e. Would it be unusual to have 0 babies out of 5 randomly selected babies with green
eyes?
30
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f. Would it be unusual to have 2 babies out of 5 randomly selected babies with green
eyes?
Find the following probabilities (assuming 5 randomly selected babies)
g. Probability of exactly 2 green eyed babies: h. Probability of 3 or more green eyed babies: i. Probability of at least 1 green eyed baby:
j. Probability of at most 1 green eyed babies:
k. Probability of no more than 1 green eyed babies: 14. Create a probability distribution for the roll of one die, where the random variable is the
value you roll. Find the mean and the standard deviation for the probability distribution.
x value
Probability P(x)
Mean Standard Deviation
15. Find the probability of getting exactly 3 correct responses among 5 different requests from
AT&T directory assistance. Assume in general, AT&T is correct 90% of the time.
n = x = p = q = Use the binomial probability formula to solve:
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16. A slot machine has a 1/2000 probability of winning the jackpot on any individual trial.
Suppose someone plays the slot machine 5 times.
a. Make a binomial probability distribution for the situation where x is defined as the
number of wins
x value
Probability P(x)
b. What is the probability of hitting the jackpot exactly twice? Is it likely?
c. What is the probability of hitting it 3 or fewer times? Is that likely?
17. Your brother baked a large batch of cookies. He put chocolate chips in 45% of the cookies.
He randomly selects 10 cookies to give to a friend. What is the probability that 6 of the
cookies contain chocolate chips?
18. How would you work each of the following problems using Technology?
60% of the class likes chocolate. If 20 students are randomly selected, find the probability: n
= p = Words
Probability Statement
Calculated Probability
Unusual (Yes or No)
a. 4 like chocolate
b. No more than 5 like chocolate
c. 7 or more like chocolate
d. More than 7 like chocolate
e. Less than 13 like chocolate
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19. What is the mean and standard deviation for the probability distribution (of 20 randomly
selected students from a class in which 60% of the students like chocolate)?
Mean: Standard Deviation: 33
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