Probability Distributions

1
Probability Distributions
ALGEBRA 2 LESSON 12-1
(For help, go to Lesson 1-6.)
Suppose you roll a standard number cube. State
whether each set represents a sample space for
the outcomes. 1. 1, 2, 3, 4, 5, 6 2. less
than 3, 4, 5, 6 3. even, prime Find each
probability for two tosses of a number
cube. 4. P(4 and 3)     5.  P(two odd
numbers)     6.  P(two integers)
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Probability Distributions
ALGEBRA 2 LESSON 12-1
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Probability Distributions
ALGEBRA 2 LESSON 12-1

Take a survey of your classmates eye colors and
make a frequency table with the data.
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Probability Distributions
ALGEBRA 2 LESSON 12-1
Use the frequency table. Find the probability
that a student is involved in at least one
extra-curricular activity.
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Probability Distributions
ALGEBRA 2 LESSON 12-1
Suppose you spin two spinners. Each spinner has
4 possible outcomes 1, 2, 3, or 4. Show the
probability distribution for the sum of the
numbers.
Method 1 Make a frequency table. Then extend
the table to include probabilities.
Method 2 Draw a graph.
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Probability Distributions
ALGEBRA 2 LESSON 12-1
Use information in the chart of inherited gene
pairs. Graph the probability distribution for
each sample space.
Inherited Gene Pairs From One Recessive and One
Hybrid Pea Plant
RR dominant gene pair (red flower) Rr hybrid
gene pair (pink flower) rr recessive gene pair
(white flower) a. Genotype Distribution b.
Plant Color Distribution Rr, rr red,
pink, white
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Probability Distributions
ALGEBRA 2 LESSON 12-1
The probability of an information desk at a
community library receiving calls C each hour
varies according to the following distribution.
Use random numbers to predict the number of calls
received during an eight-hour shift.
Step 1 Define how the simulation will be done.
Use random numbers. Assign numbers from 1 to
100 to the events based on the probability of
each event. Use cumulative probabilities to help
you assign the numbers.
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Probability Distributions
ALGEBRA 2 LESSON 12-1
(continued)
Step 2  Conduct the simulation. Model an
eight-hour period by generating eight random
numbers from 1 to 100.
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Probability Distributions
ALGEBRA 2 LESSON 12-1
(continued)
To construct the table Find the random number in
the assigned numbers column, then assign the
outcome as the number of calls. The random
number 41 is assigned to the outcome of 2 calls.
Step 3  Interpret the simulation. Based on this
simulation, a total of 19 calls came into the
information desk over an eight-hour period.
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Probability Distributions
ALGEBRA 2 LESSON 12-1
1. During lunch on Monday, the cafeteria deli
sold soft drinks to 32 customers, bottled waters
to 12 customers, sandwiches to 16 customers,
tacos to 12 customers, salads to 13
customers, and baked potatoes to 9 customers.
There were a total of 50 customers during lunch
on Monday. a.  Make a frequency table for the
data. Extend the table to include a
probability distribution. b.  What is the
probability that the next customer will
order a drink?
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Probability Distributions
ALGEBRA 2 LESSON 12-1
2. Given the data on the number of bus rides per
week for 100 people surveyed, conduct a
simulation for the number of bus rides per week
for the next ten people surveyed.
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Probability Distributions
ALGEBRA 2 LESSON 12-1
1. During lunch on Monday, the cafeteria deli
sold soft drinks to 32 customers, bottled waters
to 12 customers, sandwiches to 16 customers,
tacos to 12 customers, salads to 13
customers, and baked potatoes to 9 customers.
There were a total of 50 customers during lunch
on Monday. a.  Make a frequency table for the
data. Extend the table to include a
probability distribution. b.  What is
the probability that the next customer will
order a drink?
0.88
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Probability Distributions
ALGEBRA 2 LESSON 12-1
2. Given the data on the number of bus rides per
week for 100 people surveyed, conduct a
simulation for the number of bus rides per week
for the next ten people surveyed.
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Conditional Probability
ALGEBRA 2 LESSON 12-2
(For help, go to Lesson 9-7.)
A spinner has four equal sections that are red,
blue, green, and yellow. Find each probability
for two spins. 1. P(blue, then blue) 2. P(red,
then yellow) 3. P(not yellow, then
green) 4. P(not blue, then not red) 5. P(at
least one green) 6. P(neither spin red)
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Conditional Probability
ALGEBRA 2 LESSON 12-2
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Conditional Probability
ALGEBRA 2 LESSON 12-2
Solutions (continued)
5. Sample space for two spins P(at least
one green) P(R G, B G, G G, Y G, G R, G B, or G
Y) 6. See sample space in Exercise 5
above. P(neither spin red) P(B B, B G, B Y, G
B, G G, G Y, Y B, Y G, or Y Y)
7 16
9 16
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Conditional Probability
ALGEBRA 2 LESSON 12-2
The table shows the results of a class survey.
Find P(own a pet female)
The condition female limits the sample space to
14 possible outcomes.
Of the 14 females, 8 own a pet.
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Conditional Probability
ALGEBRA 2 LESSON 12-2
Using the data in the table, find the
probability that a sample of not recycled waste
was plastic.
The given condition limits the sample space to
non-recycled waste.
A favorable outcome is non-recycled plastic.
The probability that the non-recycled waste was
plastic is about 13.
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Conditional Probability
ALGEBRA 2 LESSON 12-2
Researchers asked people who exercise regularly
whether they jog or walk. Fifty-eight percent of
the respondents were male. Twenty percent of all
respondents were males who said they jog. Find
the probability that a male respondent jogs.
The probability that a male respondent jogs is
about 34.
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Conditional Probability
ALGEBRA 2 LESSON 12-2
Jim created the tree diagram   after examining
years of weather observations in his hometown.
The diagram shows the probability of whether a
day will begin clear or cloudy, and then the
probability of rain on days that begin clear and
cloudy.
a. Find the probability that a day will start
out clear, and then will rain.
The path containing clear and rain represents
days that start out clear and then will rain.
P(clear and rain) P(rain clear)
P(clear) 0.04 0.28 0.011
The probability that a day will start out clear
and then rain is about 1.
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Conditional Probability
ALGEBRA 2 LESSON 12-2
(continued)
b. Find the probability that it will not rain
on any given day.
The paths containing clear and no rain and cloudy
and no rain both represent a day when it will
not rain. Find the probability for both paths and
add them.
P(clear and no rain) P(cloudy and no rain)
P(clear) P(no rain clear) P(cloudy)
P(no rain cloudy) 0.28(.96) .72(.69)
0.7656
The probability that it will not rain on any
given day is about 77.
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Conditional Probability
ALGEBRA 2 LESSON 12-2
1. A study was conducted at Central High School
to find out how much television students watch
per night. Thirty-eight percent of the
respondents were freshmen. Twenty-two percent
of all respondents were freshmen who said that
they watch at least one hour of television per
night. Find the probability that a freshman
respondent watches at least one hour of
television per night. 2. The tree diagram below
shows the probabilities that Jose will or will
not cook breakfast on weekends (WE) or weekdays
(WD). Find the probability that Jose will cook
breakfast on any given day.
about 58
about 38
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Analyzing Data
ALGEBRA 2 LESSON 12-3
(For help, go to Lessons 1-1.)
Order each set of values from least to greatest.
Then find the middle value. 1. 0.2 0.3 0.6 1.2 0
.7 0.9 0.8 2. 11 23 15 17 21 18 21 3. 7.8 2.6 
3.9 15.6 9.1 11.7 10.4 4. 76 89 80 82 86 84 86
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Analyzing Data
ALGEBRA 2 LESSON 12-3
Solutions
1. 0.2, 0.3, 0.6, 0.7, 0.8, 0.9, 1.2 middle
value 0.7 2. 11, 15, 17, 18, 21, 21, 23
middle value 18 3. 2.6, 3.9, 7.8, 9.1, 10.4,
11.7, 15.6 middle value 9.1 4. 76, 80, 82,
84, 86, 86, 89 middle value 84
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Analyzing Data
ALGEBRA 2 LESSON 12-3
Find the mean, median, and mode for these
values 78, 87, 84, 75, 80, 98, 78, 95, 72.
The mean is 83, the median is 80, and the mode is
78.
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Analyzing Data
ALGEBRA 2 LESSON 12-3
Using the data in the table, find the mean,
median, and mode for the water temperatures in
Dauphin Island, AL.
Gulf of Mexico Eastern Coast Water Temperatures
(F)
Step 1 Use the STAT feature to enter data as L1
in your graphing calculator.
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Analyzing Data
ALGEBRA 2 LESSON 12-3
(continued)
Step 3 Return to the same menu to find the
median.
The mean is about 69.1F, the median is 71F, and
the mode is 84F.
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Analyzing Data
ALGEBRA 2 LESSON 12-3
Make a box-and-whisker plot for these values
91, 95, 88, 85, 90, 97, 94, 100, 81.
The minimum value is 81 and the maximum value is
100.
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Analyzing Data
ALGEBRA 2 LESSON 12-3
(continued)
Step 2  Draw a number line for the base of your
box-and-whisker plot. Above the number line,
plot the three quartiles, the minimum value,
and the maximum value.
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Analyzing Data
ALGEBRA 2 LESSON 12-3
Use a graphing calculator to find the quartiles
of the water temperature data for Dauphin, AL in
Additional Example 2.
Use the TRACE feature to find the quartile
values. They are Q1 58, Q2 71, and Q3 81.
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Analyzing Data
ALGEBRA 2 LESSON 12-3
Find the 30th and 60th percentiles for the
values below.
54 98 45 87 98 64 21 61 71 82 93 65 62 98 87 24 65
 97 31 47
Step 1  Order the values. 21 24 31 45 47 54 61 6
2 64 65 65 71 82 87 87 93 97 98 98 98
Step 2  Find the number of values that fall
below the 30th percentile and the number that
fall below the 60th percentile.
Of the 20, 30 should fall below the 30th
percentile and 60 should fall below the 60th
percentile.
20 ? 30 20 ? 0.30 6 Since 61 is greater
than 6 values, 61 is at the 30th percentile.
20 ? 60 20 ? 0.60 12 Since 82 is greater
than 12 values, 82 is at the 60th percentile.
The value at the 30th percentile is 61 and the
value at the 60th percentile is 82.
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Analyzing Data
ALGEBRA 2 LESSON 12-3
Identify an outlier for this set of values 15
34 28 32 30 26 34.
15 is substantially different, so 15 is an
outlier.
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Analyzing Data
ALGEBRA 2 LESSON 12-3
1. Marina has the following quiz grades in her
Algebra 2 class 84, 90, 96, 100, 67, 88, 90,
92, 94. a. Find the mean, median, and mode for
Marinas quiz grades. b. Make a
box-and-whisker plot for Marinas
grades. 2. Find the values at the 40th and
70th percentiles for the values below. 26, 37,
18, 45, 20, 36, 22, 25, 50, 41 3. Identify an
outlier for this set of data 65, 70, 56, 45, 59,
67, 62.
89 90 90
26 41
45
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Standard Deviation
ALGEBRA 2 LESSON 12-4
(For help, go to Skills Handbook page 845.)
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Standard Deviation
ALGEBRA 2 LESSON 12-4
Solutions
34.3 7
1. 34.3 7 4.9 2. 6 2.4
2.5 3. 8.4 1.25 10.5 4. 12 6 0.5 12
3 9 5. (2 6)2 (7 6)2 (8 6)2
(4)2 12 22 16 1 4
21 7 6. (4 3)2 (5 3)2
12 82 64 8.03
6 2.4
1 3
1 3
1 3
1 3
1 2
1 2
1 2
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Standard Deviation
ALGEBRA 2 LESSON 12-4
There are 9 members of the Community Youth
Leadership Board. Find the range and
interquartile range of their ages 22, 16, 24,
17, 16, 25, 20, 19, 26.
greatest value least value 26 16 Find the
range. 10
Q3 Q1 24.5 16.5 Find the interquartile
range. 8
The range is 10 years. The interquartile range is
8 years.
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Standard Deviation
ALGEBRA 2 LESSON 12-4
Find the mean and the standard deviation for the
values 78.2, 90.5, 98.1, 93.7, 94.5.
The mean is 91, and the standard deviation is
about 6.8.
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Standard Deviation
ALGEBRA 2 LESSON 12-4
Use the data to find the mean and standard
deviation for daily energy demands on the
weekends only.
Step 1 Use the STAT feature to enter the data
as L1.
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Standard Deviation
ALGEBRA 2 LESSON 12-4
The number of points that Darden scored in each
of 11 basketball games is listed below. Within
how many standard deviations of the mean do all
of the values fall? What can Dardens coach do
with this information?
8, 12, 13, 10, 7, 5, 10, 9, 13, 11, 8
All the values fall within 2 standard deviations
of the mean.
The coach can expect that it will be very
likely that his score in the next game will be
within 5 points of his mean score of 10 points.
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Standard Deviation
ALGEBRA 2 LESSON 12-4
A set of values has a mean of 22 and a standard
deviation of 3. Find the z-score for a value of
24.
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Standard Deviation
ALGEBRA 2 LESSON 12-4
Use the following set of data for Questions 1
through 3 below. 13, 10, 11, 8, 14, 13, 13, 14,
30 1. Find the range and interquartile range
for the set of data. 2. Find the mean and
standard deviation for the set of
data. 3. Within how many standard deviations of
the mean do all of the data values
fall? 4. A set of values has a mean of 34 and a
standard deviation of 4. Find the z-score of
the value 26.
22 3.5
14 about 6.0
3
2
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Working With Samples
ALGEBRA 2 LESSON 12-5
(For help, go to Lesson 7-1.)
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Working With Samples
ALGEBRA 2 LESSON 12-5
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Working With Samples
ALGEBRA 2 LESSON 12-5
In a sample of 500 teenagers, 328 had never
attended a popular music concert. Find the sample
proportion for those who have never attended a
concert. Write the answer as a percent.
0.66 Simplify.
The sample proportion of teenagers who have never
attended a popular music concert is about 66.
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Working With Samples
ALGEBRA 2 LESSON 12-5
The Sunnyvale High School student council dance
committee is trying to decide whether to have a
band or a DJ for the fall dance. They decided
that each of the four committee members should
survey the students in their homeroom classes.
Identify any bias in this sampling method.
This is a convenience sample that is convenient
for the committee members.
Four homerooms may not accurately reflect the
opinions of the entire school, because four
homerooms is probably a low percentage of all
the schools homerooms.
This sampling method has a bias and is not random.
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Working With Samples
ALGEBRA 2 LESSON 12-5
In a survey, teenagers were asked to rank the
importance of their relationships with their
parents. The response scale ranged from 1 to 5,
with 5 being extremely important. Use the
information in the table to decide which sample
was most likely the greatest in size.
Sample B was most likely the greatest in size
since it has the smallest standard deviation.
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Working With Samples
ALGEBRA 2 LESSON 12-5
An opinion poll about the popularity of the
mayor has a margin of error of 5. Estimate the
number of people who were surveyed.
20 Simplify.
n 400 Square each side.
The poll surveyed about 400 people.
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Working With Samples
ALGEBRA 2 LESSON 12-5
A survey of 528 high school seniors found that
65 already had career plans after high school.
a. Find the margin of error for the sample.
The margin of error is about 4.
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Working With Samples
ALGEBRA 2 LESSON 12-5
(continued)
b. Use the margin of error to find an interval
that is likely to contain the true population
proportion.
The proportion of seniors who already have career
plans is likely to be from 61 to 69.
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Working With Samples
ALGEBRA 2 LESSON 12-5
1. If 412 of 1720 students at a high school
participate in extra-curricular sports
activities, what is the proportion of the
students who participate in sports? Write the
proportion as a percent. 2. A radio talk show
found that over 90 of listeners who called in
were against building a community recreation
center. Identify any bias in this sampling
method. 3. A biology class counted the number
of microorganisms in drops of water from a
nearby pond. The class took three samples. Use
the information in the table to decide which
sample was largest. Explain your reasoning.
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Working With Samples
ALGEBRA 2 LESSON 12-5
4. A survey of voters found that 61 of voters
supported building a new high school, with a
margin of error of 2. Estimate the number of
voters in the poll. 5. Jill conducted a
survey of 100 drivers and found that 72 thought
that improperly timed traffic lights were the
citys worst traffic problem. Find the margin
of error and the interval that is likely to
contain the true population proportion.
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Working With Samples
ALGEBRA 2 LESSON 12-5
1. If 412 of 1720 students at a high school
participate in extra-curricular sports
activities, what is the proportion of the
students who participate in sports? Write the
proportion as a percent. 2. A radio talk show
found that over 90 of listeners who called in
were against building a community recreation
center. Identify any bias in this sampling
method. 3. A biology class counted the number
of microorganisms in drops of water from a
nearby pond. The class took three samples. Use
the information in the table to decide which
sample was largest. Explain your reasoning.
about 24
This is a self-selected sample, and the people
who call in are likely to over-
or under-represent certain viewpoints.
Sample A Sample A has the lowest standard
deviation suggesting that is was the largest
sample.
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Working With Samples
ALGEBRA 2 LESSON 12-5
4. A survey of voters found that 61 of voters
supported building a new high school, with a
margin of error of 2. Estimate the number of
voters in the poll. 5. Jill conducted a
survey of 100 drivers and found that 72 thought
that improperly timed traffic lights were the
citys worst traffic problem. Find the margin
of error and the interval that is likely to
contain the true population proportion.
about 2500 voters
10 6282
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Binomial Distributions
ALGEBRA 2 LESSON 12-6
(For help, go to Lessons 6-7 and 6-8.)
Evaluate each expression. 1. 4C2 2. 3C3 3. 5C2
Use the binomial theorem to expand each
binomial. 4. (x 2)3 5. (w y)4 6. (m
n)3 7. (t 3s)4 8. (a 2b)5 9. (p q)6
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Binomial Distributions
ALGEBRA 2 LESSON 12-6
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Binomial Distributions
ALGEBRA 2 LESSON 12-6

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Binomial Distributions
ALGEBRA 2 LESSON 12-6
Suppose you and a friend flip a coin three times
to find out who gets to play an arcade game
first. The person who gets two out of three flips
wins. Your winning side is heads.
a. Describe a trial for this situation. How many
trials are there?
Each flip is a trial. Since there are three
flips, there are three trials.
b. Describe a success. What is the probability of
a success on each trial?
Heads is a success. Since there are two possible
outcomes, each equally likely, the probability
of success on any single trial is 0.5.
c. Conduct a simulation to find the probability
of getting heads two out of three times. Run
the simulation ten times.
Assign each face of the coin, heads and tails, to
an outcome, based on each outcomes probability.
Let 1 represent heads and 2 represent tails. Use
a graphing calculator or a random number table to
generate random numbers from 1 to 2.
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Binomial Distributions
ALGEBRA 2 LESSON 12-6
(continued)
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Binomial Distributions
ALGEBRA 2 LESSON 12-6
A fast food restaurant is attaching prize cards
to every one of its soft drink cups. The
restaurant awards free drinks as prizes on three
out of four cards. Suppose you have three cards.
Find the probability that exactly one of these
cards will reveal a prize.
The probability that exactly one of three cards
will reveal a free drink is about 14.
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Binomial Distributions
ALGEBRA 2 LESSON 12-6
Alicia walks to school with her friend Juana.
Juana is on time 80 of the time. What is the
probability that Juana will be on time five days
in a row?
Relate  This is a binomial experiment. There
are five days. Each day shell be on time or
late. The probability of on time is 0.8 for
each day.
Define Let n 5. Let p 0.8. Let q
0.2. Let x 5.
Write nCxpxqnx 5C5(0.8)5(0.2)0 Substitute.
(1)(0.8)5(1) Simplify.
0.32768 Simplify.
The probability that she will be on time five
days in a row is about 33.
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Binomial Distributions
ALGEBRA 2 LESSON 12-6
When it rains, there is a 70 chance that
Malcolms soccer practice will be cancelled. If
it rains for the next three days, what is the
probability that Malcolms practice will be
cancelled on at least one of the days?
Use the expansion for (p q)n, with n 3, p
0.7, and q 0.3.
P(at least 1 success) P(1 success) P(2
successes) P(3 successes)
0.189 0.441 0.343
0.973
The probability that at least one practice will
be cancelled if it rains for the next three days
is about 97.
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Binomial Distributions
ALGEBRA 2 LESSON 12-6
1. North High Schools football team has a
reputation for losing the first game of every
season. In the last 20 years, they have lost 70
of their opening games. What is the probability
that they will lose their season opening games
for the next three years? 2. Yolanda enjoys
fishing even though she only catches fish on
about 20 of the days she spends fishing. If
Yolanda is going on a 4-day fishing trip, what is
the probability that she will catch fish on at
least one of the four days?
about 34
about 59
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Normal Distributions
ALGEBRA 2 LESSON 12-7
(For help, go to Lesson 12-4.)
Find the numbers that are one and two standard
deviations above and below each given
mean. 1. x 12 ? 2 2. x 16.7 ?
1 3. x 7 ? 1.5 4. x 22 ? 1.7 5. x
17.5 ? 0.9 6. x 33.1 ? 1.2
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Normal Distributions
ALGEBRA 2 LESSON 12-7
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Normal Distributions
ALGEBRA 2 LESSON 12-7
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Normal Distributions
ALGEBRA 2 LESSON 12-7
Use the data from Example 1 to estimate the
number of babies that are between 1.5 and 2.5
standard deviations from the mean birth weight.
The standard deviation in birth weights is about
500g.
Estimate and add the percents for 20002499 and
40004499.
5 5 10
About 10 of the babies are between 1.5 and 2.5
standard deviations from the mean birth weight.
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Normal Distributions
ALGEBRA 2 LESSON 12-7
A survey of the employees of XYZ Corporation
found that the mean of the morning commute times
to work was 18 minutes. The standard deviation
was 4 minutes. Sketch a normal curve showing the
commute times at one, two, and three standard
deviations from the mean.
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Normal Distributions
ALGEBRA 2 LESSON 12-7
In a survey, the responses to the question, How
much time do you spend in the shower every day?
were normally distributed. The mean was 15
minutes the standard deviation was 2 minutes.
a. What values are one deviation from the mean?
b. What percent of the responses would you
expect to find that are less than 13 and
greater than 17?
Values that are one standard deviation from the
mean have z-scores of 1 and 1.
The responses are normally distributed, and 13
and 17 are the values that are one standard
deviation from the mean.
Since 68 of the data are within one standard
deviation of the mean, 100 68 32 should
be the values outside one standard deviation of
the mean, i.e. values less than 13 and greater
than 17.
The values 13 minutes and 17 minutes are one
standard deviation from the mean.
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Normal Distributions
ALGEBRA 2 LESSON 12-7
Use the distribution to answer the following
questions. Assume there are 92 students in the
lecture class.
The professor has found that students who score
between one and two standard deviations below
the mean need to attend study sessions in order
to pass the class. How many students need to
attend study sessions?
Use the normal curve. About 13.5 of the students
receive grades from one to two standard
deviations below the mean.
Find the number of students that corresponds to
13.5.
0.135(92) 12.42
About 12 or 13 students need to attend study
sessions.
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Normal Distributions
ALGEBRA 2 LESSON 12-7
1. The bar graph below gives the hours of
homework per week of surveyed students.
a.  About what percent of students have
between 5 and 9 hours of homework per
week? b.  Suppose the survey included 500
students. About how many have between 5 and 9
hours of homework per week? c.  The mean of the
data is seven hours and the standard deviation is
two. About what percentage of students have an
amount of homework within 2 standard
deviations of the mean?
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Normal Distributions
ALGEBRA 2 LESSON 12-7
2. A biology class did a survey of an open field
for the number of visible insects per square
foot of ground. The results were normally
distributed. The mean was 28 insects and the
standard deviation was 5. a.  What values are
1 standard deviation from the mean? b.  Suppose
the class surveyed 50 one-foot square plots.
Find the number of plots that contained a
number of insects within one standard
deviation of the mean.
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Normal Distributions
ALGEBRA 2 LESSON 12-7
1. The bar graph below gives the hours of
homework per week of surveyed students.
a.  About what percent of students have
between 5 and 9 hours of homework per
week? b.  Suppose the survey included 500
students. About how many have between 5 and 9
hours of homework per week? c.  The mean of the
data is seven hours and the standard deviation is
two. About what percentage of students have an
amount of homework within 2 standard
deviations of the mean?
about 66
about 330 students
about 92
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Normal Distributions
ALGEBRA 2 LESSON 12-7
2. A biology class did a survey of an open field
for the number of visible insects per square
foot of ground. The results were normally
distributed. The mean was 28 insects and the
standard deviation was 5. a.  What values are
1 standard deviation from the mean? b.  Suppose
the class surveyed 50 one-foot square plots.
Find the number of plots that contained a
number of insects within one standard
deviation of the mean.
23 and 33 insects
34 plots
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